#72343270

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 ◁ #20385 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3069 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:

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

Polycyclic presentation:

2^10|a^2=f,f^2=j,h^2=i,i^2=j,b^a=bi,c^b=cg,d^a=dh,d^b=dh,d^c=dgj,e^b=egj,f^d=fj,h^a=hi,h^b=hij,h^d=hij,i^a=ij,i^b=ij,i^d=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343283
maximal normal subgroup of	#72343283
maximal quotient of	#72343805

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