#72265035

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×#102
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	#10576590	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#525 ◁ C₂×#2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

5447 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^10|b^2=f,c^2=fghi,g^2=i,i^2=j,c^a=ch,c^b=cg,d^c=dj,e^a=eh,e^d=ej,f^c=fj,g^b=gi,g^c=gj,i^b=ij

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265073, #72265164, #72308595
maximal normal subgroup of	#72265073, #72265164, #72308595

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