#6753016

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1591 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6660212);