#72265116

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

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	C₂×#3155	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ #20331 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3337 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)(3,7)(4,6)(5,8)(11,19,22,32)(12,23)(13,26)(16,20,28,35)(17,24)(18,27)(21,31,37,33)(25,38)(29,39)(30,34,40,36);(1,3)(2,5)(4,8)(6,7)(9,10)(11,20,22,35)(12,24)(13,27)(14,15)(16,19,28,32)(17,23)(18,26)(21,34,37,36)(25,39)(29,38)(30,31,40,33);(11,21,22,37)(12,25)(16,30,28,40)(17,29)(19,31,32,33)(20,34,35,36)(23,38)(24,39);(10,15)(11,22)(17,29)(18,27)(19,32)(21,37)(24,39)(31,33);(2,6)(3,8)(9,11,12,19,13,21,23,33,14,22,25,32,26,37,38,31)(10,16,17,20,18,30,24,36,15,28,29,35,27,40,39,34)

Polycyclic presentation:

2^10|a^2=f,b^2=f,c^2=f,e^2=gh,g^2=i,i^2=j,b^a=bh,d^b=dj,e^a=eg,e^b=eg,e^c=ei,e^d=ej,f^e=fj,g^a=gi,g^b=gi,g^c=gj,i^a=ij,i^b=ij

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265164, #72308189
maximal normal subgroup of	#72265164, #72308189

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