#6883866

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	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1508 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)(20,25,35,43)(21,26,22,37)(24,30,39,31)(29,44,33,48)(34,41,36,42)(38,47,40,46);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14);(2,6)(3,8)(10,14)(15,16)(17,19,22,31,23,32,21,30)(18,24,26,28,27,39,37,45)(20,33,34,47,35,29,36,46)(25,40,42,48,43,38,41,44);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,20)(18,25)(19,29)(21,36)(22,34)(23,35)(24,38)(26,42)(27,43)(28,44)(30,47)(31,46)(32,33)(37,41)(39,40)(45,48);(9,12,13,11)(10,15,14,16)(17,21,23,22)(18,26,27,37)(19,30,32,31)(20,34,35,36)(24,28,39,45)(25,41,43,42)(29,46,33,47)(38,48,40,44)

Polycyclic presentation:

2^9|a^2=i,c^2=fg,d^2=fh,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,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, 6791062);