#6757165

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1775 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6664361);