#6814184

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	#42664	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1429 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)(4,8)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,21,25)(19,27,30,23)(20,24,33,39)(22,26,34,40)(28,41,43,35)(29,38,32,36)(31,37,44,47)(42,48,45,46);(1,3,6,2)(4,8,5,7)(9,11,12,15)(10,13,14,16)(19,28)(20,31)(22,34)(23,35)(24,37)(26,40)(27,41)(29,32)(30,43)(33,44)(36,38)(39,47);(10,14)(13,16)(17,19,22,28,21,30,34,43)(18,23,26,35,25,27,40,41)(20,32,31,45,33,29,44,42)(24,38,37,48,39,36,47,46);(1,4,6,5)(2,7,3,8)(20,33)(24,39)(29,32)(31,44)(36,38)(37,47)(42,45)(46,48);(1,5,6,4)(2,7,3,8)(17,20,22,31,21,33,34,44)(18,24,26,37,25,39,40,47)(19,29,28,42,30,32,43,45)(23,36,35,46,27,38,41,48)

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=h,d^2=f,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,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, 6721380);