#72235260

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#238077
commutator, Frattini, cosocle	C₂²×#121	C₂⁴
prime radical	#5025789	C₂²
LCS	G ▷ C₂²×#121 ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#560 ◁ G
LECS	G ▷ C₂²×#121 ▷ C₂⁴ ▷ C₂² ▷ C₁

3088 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,6,13)(3,18)(4,24,28,60)(5,32)(7,14,38,54)(8,40)(9,17,36,16)(10,22)(11,48,50,19)(12,34)(15,42)(20,58,29,53)(21,44,37,51)(23,46)(25,52,33,55)(26,64,63,49)(27,45,41,57)(30,61,31,62)(35,43,39,56)(47,59);(1,3,9,22)(2,10,13,47)(4,25,31,63)(5,21,42,48)(6,23,36,59)(7,29)(8,37,34,19)(11,40)(12,51)(14,53,54,45)(15,44)(16,46,17,18)(20,39)(24,55,61,52)(26,30,33,28)(27,38)(32,50)(35,41)(43,57,56,58)(49,62,64,60);(1,4,26,59,36,28,25,3)(2,11,49,45,16,51,55,53)(5,29,7,37,42,27,38,19)(6,31,33,22,9,30,63,23)(8,41,35,21,34,20,39,48)(10,43,62,32,46,56,61,15)(12,47,14,24,40,18,54,60)(13,50,52,58,17,44,64,57);(1,5,9,34)(2,12,17,32)(3,19,22,21)(4,27,30,41)(6,8,36,42)(7,33,39,25)(10,44,18,11)(13,15,16,40)(14,52,56,55)(20,28,29,31)(23,48,59,37)(24,45,61,57)(26,38,63,35)(43,49,54,64)(46,50,47,51)(53,62,58,60)

Polycyclic presentation:

2^11|a^2=e,b^2=eij,c^2=efgj,d^2=j,f^2=jk,b^a=behik,c^a=cf,c^b=cfg,d^a=dhijk,d^b=dij,d^c=dhk,e^b=ek,e^c=eh,e^d=ek,f^a=fhjk,f^b=fhi,f^c=fijk,f^e=fj,g^a=ghijk,g^b=gik,g^c=ghijk,g^d=gj,g^e=gk,g^f=gjk,h^a=hj,h^b=hjk,h^c=hjk,i^a=ijk,i^b=ik,i^c=ik

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72235272
maximal normal subgroup of	#72235272

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.5*F.9*F.10);;
gap> SetPower(coll, 3, F.5*F.6*F.7*F.10);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 6, F.10*F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.5*F.8*F.9*F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 4, 2, F.4*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.11);;
gap> SetConjugate(coll, 5, 2, F.5*F.11);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.10*F.11);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10*F.11);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.11);;
gap> SetConjugate(coll, 7, 6, F.7*F.10*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10*F.11);;
gap> SetConjugate(coll, 8, 3, F.8*F.10*F.11);;
gap> SetConjugate(coll, 9, 1, F.9*F.10*F.11);;
gap> SetConjugate(coll, 9, 2, F.9*F.11);;
gap> SetConjugate(coll, 9, 3, F.9*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);