#6696925

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1586 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)(3,6)(4,5)(7,8)(10,14)(11,12)(17,18,22,26)(19,23,30,39)(20,24,32,41)(21,25,34,43)(27,36,29,38)(28,37,46,48)(31,40,45,47)(33,42,35,44);(9,10)(11,15)(12,16)(13,14)(17,19,22,30)(18,23,26,39)(20,31,32,45)(21,27,34,29)(24,40,41,47)(25,36,43,38)(28,35,46,33)(37,44,48,42);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(19,27)(20,32)(21,34)(23,36)(24,41)(25,43)(28,45)(29,30)(31,46)(37,47)(38,39)(40,48);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,14,15)(20,32)(24,41)(28,46)(31,45)(33,35)(37,48)(40,47)(42,44);(10,14)(15,16)(17,20,22,32)(18,24,26,41)(19,28,30,46)(21,35,34,33)(23,37,39,48)(25,44,43,42)(27,31,29,45)(36,40,38,47)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6604121);