#6680128

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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×#555	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1464 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,27)(19,24,32,41)(20,25,35,43)(21,26,37,22)(28,39,46,47)(29,31,40,30)(33,42,45,48)(34,36,44,38);(2,5)(3,7)(9,11,12,15)(10,13,14,16)(17,19,23,32)(18,24,27,41)(20,33,35,45)(21,29,37,40)(22,30,26,31)(25,42,43,48)(28,38,46,36)(34,47,44,39);(10,14)(13,16)(17,20,23,35)(18,25,27,43)(19,28,32,46)(21,34,37,44)(22,38,26,36)(24,39,41,47)(29,42,40,48)(30,33,31,45);(2,5)(6,8)(17,21)(18,26)(19,29)(20,34)(22,27)(23,37)(24,31)(25,36)(28,42)(30,41)(32,40)(33,47)(35,44)(38,43)(39,45)(46,48);(1,3,4,7)(2,6,5,8)(19,30)(20,35)(21,37)(22,26)(24,29)(25,43)(28,45)(31,32)(33,46)(39,48)(40,41)(42,47)

Polycyclic presentation:

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

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, 6587324);