#6791313

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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 ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1678 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)(17,18)(19,24)(20,30)(21,26)(22,36)(23,27)(25,28)(29,38)(31,40)(32,43)(33,37)(34,41)(35,46)(39,42)(44,48)(45,47);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,13)(14,16)(17,19,23,22)(18,24,27,36)(20,31,33,28)(21,29,34,35)(25,30,40,37)(26,38,41,46)(32,44,45,42)(39,43,48,47);(10,13)(14,15)(17,20,23,33)(18,25,27,40)(19,28,22,31)(21,32,34,45)(24,37,36,30)(26,39,41,48)(29,42,35,44)(38,47,46,43);(2,6)(5,7)(17,21)(18,26)(19,29)(20,32)(22,35)(23,34)(24,38)(25,39)(27,41)(28,42)(30,43)(31,44)(33,45)(36,46)(37,47)(40,48);(3,8)(5,7)(9,11,12,16)(10,14,13,15)(21,34)(26,41)(29,35)(32,45)(38,46)(39,48)(42,44)(43,47)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6698509);