#6692485

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1448 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)(9,10,13,15)(11,14,12,16)(17,18)(19,23)(20,25)(21,24)(22,26)(27,43)(28,37)(29,38)(30,36)(31,41)(32,39)(33,42)(34,40)(35,44)(45,48)(46,47);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,15,16)(17,19,21,29)(18,23,24,38)(20,31,32,46)(22,27,34,30)(25,41,39,47)(26,43,40,36)(28,35,45,33)(37,44,48,42);(9,12,13,11)(10,14,15,16)(18,24)(19,27)(22,34)(23,36)(25,39)(28,31)(29,30)(33,35)(37,47)(38,43)(41,48)(45,46);(2,6)(5,7)(20,32)(25,39)(28,45)(31,46)(33,35)(37,48)(41,47)(42,44);(3,8)(5,7)(17,20,21,32)(18,25,24,39)(19,28,29,45)(22,35,34,33)(23,37,38,48)(26,44,40,42)(27,31,30,46)(36,47,43,41)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,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, 6599681);