#72343221

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#392828
commutator	C₂×#131	C₂³×C₄
Frattini, cosocle	C₄×#131	C₂⁴
socle	C₂	#72343141
prime radical	#6469078	C₂²
LCS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×#131 ◁ G
LECS	G ▷ C₄×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

2203 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:

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

Polycyclic presentation:

2^11|a^2=e,b^2=h,d^2=g,e^2=i,h^2=j,i^2=k,j^2=k,b^a=bj,c^a=ck,c^b=cijk,d^a=dg,d^b=dghi,d^c=df,e^b=ek,f^a=fijk,f^b=fij,f^d=fij,g^c=gij,g^f=gk,h^a=hk,h^d=hj,j^d=jk

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343240
maximal normal subgroup of	#72343240

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.8);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.11);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.11);;
gap> SetConjugate(coll, 3, 2, F.3*F.9*F.10*F.11);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.6);;
gap> SetConjugate(coll, 5, 2, F.5*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.9*F.10*F.11);;
gap> SetConjugate(coll, 6, 2, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.11);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 10, 4, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);