#6818862

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1317 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)(11,14)(12,16)(13,15)(17,18,21,25)(19,27,30,23)(20,31,33,24)(22,26,34,38)(28,39,42,35)(29,36,43,40)(32,44,45,37)(41,46,48,47);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,15,16)(19,28)(20,32)(22,34)(23,35)(24,37)(26,38)(27,39)(29,43)(30,42)(31,44)(33,45)(36,40);(9,12,13,11)(10,14,15,16)(17,19,22,28,21,30,34,42)(18,23,26,35,25,27,38,39)(20,29,32,41,33,43,45,48)(24,36,37,46,31,40,44,47);(2,6)(5,7)(20,33)(24,31)(29,43)(32,45)(36,40)(37,44)(41,48)(46,47);(3,8)(5,7)(17,20,22,32,21,33,34,45)(18,24,26,37,25,31,38,44)(19,29,28,41,30,43,42,48)(23,36,35,46,27,40,39,47)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=ghi,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, 6726058);