#72235267

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#119556
commutator	C₂²×#123	C₂²×C₄
Frattini, cosocle	C₂²×#560	C₂³
prime radical	#5025789	C₂²
LCS	G ▷ C₂²×#123 ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#560 ◁ G
LECS	G ▷ C₂²×#560 ▷ C₂⁴ ▷ C₂² ▷ C₁

2758 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^11|a^2=d,b^2=ei,c^2=de,g^2=j,b^a=bde,c^a=cef,c^b=cdefgjk,d^b=di,d^c=dh,e^a=eijk,e^b=ejk,e^c=eh,e^d=ejk,f^a=fhi,f^b=fij,f^c=fhi,f^d=fk,f^e=fjk,g^a=ghi,g^b=ghj,g^c=ghk,g^d=gk,g^e=gj,g^f=gj,h^a=hj,h^b=hj,h^c=hjk,i^a=ijk,i^b=ijk,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.4);;
gap> SetPower(coll, 2, F.5*F.9);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.4*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.5*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.4*F.5*F.6*F.7*F.10*F.11);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.9*F.10*F.11);;
gap> SetConjugate(coll, 5, 2, F.5*F.10*F.11);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.10*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.11);;
gap> SetConjugate(coll, 6, 5, F.6*F.10*F.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.8*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.8*F.11);;
gap> SetConjugate(coll, 7, 4, F.7*F.11);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
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.10*F.11);;
gap> SetConjugate(coll, 9, 3, F.9*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);