#6680290

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
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

1471 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)(17,18,23,21)(19,24,31,28)(20,25,34,35)(22,26,38,36)(27,39,45,46)(29,40,30,41)(32,42,47,48)(33,43,37,44);(2,5)(3,7)(9,11,12,15)(10,13,14,16)(17,19,23,31)(18,24,21,28)(20,32,34,47)(22,29,38,30)(25,42,35,48)(26,40,36,41)(27,37,45,33)(39,44,46,43);(10,14)(13,16)(17,20,23,34)(18,25,21,35)(19,27,31,45)(22,37,38,33)(24,39,28,46)(26,44,36,43)(29,32,30,47)(40,42,41,48);(2,5)(6,8)(17,21,23,18)(19,28,31,24)(20,25,34,35)(22,36,38,26)(27,39,45,46)(29,41,30,40)(32,42,47,48)(33,43,37,44);(1,3,4,7)(2,6,5,8)(19,29)(22,38)(24,40)(26,36)(27,32)(28,41)(30,31)(33,37)(39,42)(43,44)(45,47)(46,48)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=i,d^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,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6587486);