#72309482

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#23029
commutator	C₄×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	#3147	C₂³
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #8203 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

860 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

(9,10,12,15)(11,19)(13,25,21,38)(14,27,24,18)(16,30)(17,32,29,26)(20,33)(22,28)(23,34)(31,36)(35,37)(39,40);(2,5)(3,7)(6,8)(11,20,23,37)(13,21)(16,31,28,40)(17,29)(19,33,34,35)(22,39,30,36)(25,38)(26,32);(1,2,4,6)(3,8,7,5)(10,15)(16,28)(17,29)(18,27)(19,34)(25,38)(31,40)(33,35);(2,6)(5,8)(9,11,21,30,14,20,32,39,12,23,13,22,24,37,26,36)(10,16,29,34,18,31,38,33,15,28,17,19,27,40,25,35)

Polycyclic presentation:

2^10|a^2=e,b^2=ej,c^2=h,d^2=gj,f^2=h,g^2=i,i^2=j,c^a=cj,c^b=cf,d^a=dg,d^b=di,d^c=dh,e^d=ej,f^b=fh,f^c=fh,g^a=gi,g^b=gj,i^a=ij

Conjugacy classes & character table

79 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309511, #72309744
maximal normal subgroup of	#72309511, #72309744

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