#6804290

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1296 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)(17,18,23,27)(19,28,32,45)(20,25,33,40)(21,26,22,37)(24,30,39,31)(29,44,34,48)(35,42,36,43)(38,47,41,46);(1,3)(2,5)(4,8)(6,7)(9,10,12,13)(11,15,16,14)(20,33)(25,40)(29,34)(35,36)(38,41)(42,43)(44,48)(46,47);(10,13)(14,15)(17,19,22,31,23,32,21,30)(18,24,26,28,27,39,37,45)(20,34,35,47,33,29,36,46)(25,41,43,48,40,38,42,44);(2,6)(5,7)(17,20,23,33)(18,25,27,40)(19,29,32,34)(21,36,22,35)(24,38,39,41)(26,43,37,42)(28,44,45,48)(30,47,31,46);(3,8)(5,7)(9,11,12,16)(10,14,13,15)(17,21,23,22)(18,26,27,37)(19,30,32,31)(20,35,33,36)(24,28,39,45)(25,42,40,43)(29,46,34,47)(38,48,41,44)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=g,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di,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, 6711486);