#72307758

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#1740
commutator	C₂³×C₄	C₂×C₄²
Frattini, prime radical, cosocle	C₂³×#41	C₂³
socle	C₂²	#10846
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ C₂³×#131 ◁ G
LECS	G ▷ C₂³×#41 ▷ C₂³ ▷ C₂ ▷ C₁

2611 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:

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

Polycyclic presentation:

2^10|a^2=e,b^2=f,c^2=g,g^2=j,b^a=bd,c^a=cg,c^b=cj,d^a=di,d^b=dh,e^b=ei,e^d=ej,f^a=fh,g^a=gj,i^a=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307765, #72307805, #72308110
maximal normal subgroup of	#72307765, #72307805, #72308110

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