#6883838

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
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)(17,18,22,27)(19,28,32,38)(20,24,35,41)(21,25,29,26)(23,36,39,31)(30,43,46,48)(33,40,34,42)(37,44,47,45);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14);(2,6)(3,8)(10,14)(15,16)(17,19,29,36,22,32,21,31)(18,23,25,38,27,39,26,28)(20,30,33,44,35,46,34,45)(24,37,42,48,41,47,40,43);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,20)(18,24)(19,30)(21,34)(22,35)(23,37)(25,42)(26,40)(27,41)(28,43)(29,33)(31,45)(32,46)(36,44)(38,48)(39,47);(9,12,13,11)(10,15,14,16)(17,21,22,29)(18,25,27,26)(19,31,32,36)(20,33,35,34)(23,38,39,28)(24,40,41,42)(30,44,46,45)(37,43,47,48)

Polycyclic presentation:

2^9|a^2=i,c^2=fgi,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,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, 6791034);