#72231965

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×C₄²
commutator	C₂×C₄	C₂²×C₄×C₈
Frattini, cosocle	C₄×C₁₆	C₂⁴
socle	C₂²	C₂²×#346
prime radical	C₂×#41	M₃₂
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

653 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 48 points:

(3,9)(7,12)(10,15)(14,16)(17,18,20,25,21,27,34,41,22,28,32,42,33,45,48,46,23,24,35,38,36,26,43,40,37,39,19,29,30,44,31,47);(1,2)(3,7)(4,8)(5,6)(9,12)(10,14)(11,13)(15,16)(18,24)(25,38)(26,27)(28,39)(29,42)(40,41)(44,45)(46,47);(2,6)(7,12)(8,13)(14,16)(18,24)(25,38)(26,27)(28,39)(29,42)(40,41)(44,45)(46,47);(1,3,4,10,5,9,11,15)(2,7,8,14,6,12,13,16)(17,19,21,31,22,20,33,34,23,32,36,48,37,35,30,43)(18,25,26,40,28,42,44,47,24,38,27,41,39,29,45,46)

Polycyclic presentation:

2^10|a^2=e,d^2=f,e^2=g,f^2=h,g^2=i,h^2=j,i^2=j,b^a=bj,c^a=cj,c^b=chij,d^a=dhj,e^d=ej,f^a=fj

Conjugacy classes & character table

220 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72231966, #72231967, #72231969
maximal normal subgroup of	#72231966, #72231967, #72231969

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