#72318622

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#32899
commutator, Frattini, cosocle	C₂³×D₄	C₂⁴
prime radical	#8824175	C₂
LCS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ #2309 ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

3798 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,10)(4,8)(5,7)(6,9)(11,14)(12,16)(13,15)(18,23)(19,26)(20,27)(24,28)(25,32)(29,30);(1,3)(2,7)(4,11)(5,6)(8,15)(9,10)(12,13)(14,16)(17,18)(19,27)(20,26)(21,25)(22,23)(24,29)(28,30)(31,32);(1,4,6,12)(2,8,9,16)(3,11,5,13)(7,14,10,15)(17,19)(18,20)(21,26)(22,24)(23,30)(25,29)(27,32)(28,31);(1,5,6,3)(2,7)(4,11,12,13)(8,14)(9,10)(15,16)(17,20)(18,24,23,19)(21,27)(22,30)(25,26,32,28)(29,31)

Polycyclic presentation:

2^10|c^2=hij,d^2=efgij,g^2=j,b^a=bhj,c^a=ce,c^b=cef,d^a=deh,d^b=defh,d^c=defgj,e^b=eh,e^d=ehi,f^a=fh,f^b=fhj,f^c=fj,f^d=fhij,f^e=fj,g^a=gij,g^c=gj,g^e=gj,g^f=gj,h^c=hj,h^d=hj,i^c=ij,i^d=ij

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318703
maximal normal subgroup of	#72318703

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