#72314304

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	#7623305	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ C₂×#1752 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2975 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^10|a^2=f,c^2=fgij,d^2=f,e^2=fhj,g^2=i,i^2=j,b^a=bh,c^a=cij,c^b=cj,d^b=dh,d^c=dgi,e^a=eh,e^b=eh,e^c=eij,f^c=fj,g^a=gj,g^c=gj,g^d=gi,g^e=gj,i^d=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314329, #72314441, #72314559
maximal normal subgroup of	#72314329, #72314441, #72314559

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