#6823461

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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	#42760	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1408 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:

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

Polycyclic presentation:

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

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, 6730657);