#6856472

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1494 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fhi,c^2=fh,d^2=i,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6763668);