#72340668

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#1390
commutator	C₂×C₄⋊C₄	C₂×C₄²
Frattini, cosocle	#2297	C₂³
prime radical	C₂³×#41	D₄
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂⁴×C₄ ◁ C₂×#2297 ◁ G
LECS	G ▷ #2297 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

1933 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^10|a^2=f,c^2=dehi,d^2=h,e^2=hi,g^2=j,c^a=cd,c^b=cj,d^a=dij,d^c=dg,e^a=eghij,e^c=eg,e^d=eij,f^c=fhij,f^d=fj,g^c=gi,g^d=gj,i^a=ij,i^c=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340669, #72340677, #72340732
maximal normal subgroup of	#72340669, #72340677, #72340732

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