#72307353

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#17595
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂×#97	C₂⁴
prime radical	C₂×#3184	C₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #20331 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1447 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^10|a^2=gj,b^2=f,c^2=gj,e^2=h,f^2=i,h^2=j,i^2=j,b^a=be,c^b=cij,d^a=dgj,e^a=eh,e^b=eg,e^c=ej,e^d=ej,f^a=fgh,f^c=fj,f^e=fj,g^b=gj,h^a=hj,i^a=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307374, #72307671, #72307699
maximal normal subgroup of	#72307374, #72307671, #72307699

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