#6826628

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1258 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)(11,15)(12,14)(13,16)(17,18)(19,30)(20,25)(21,29)(22,37)(23,28)(24,32)(26,34)(27,39)(31,40)(33,42)(35,44)(36,43)(38,41)(45,48)(46,47);(9,11)(10,13)(12,16)(14,15)(17,19,23,32)(18,24,28,30)(20,34,33,21)(22,35,39,38)(25,29,42,26)(27,44,37,41)(31,36,47,48)(40,45,46,43);(2,5)(6,8)(11,16)(13,15)(17,20,21,19,23,33,34,32)(18,25,29,30,28,42,26,24)(22,38,36,47,39,35,48,31)(27,44,45,40,37,41,43,46);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,21,23,34)(18,26,28,29)(19,20,32,33)(22,36,39,48)(24,25,30,42)(27,43,37,45)(31,35,47,38)(40,41,46,44);(17,22,21,36,23,39,34,48)(18,27,29,45,28,37,26,43)(19,31,33,38,32,47,20,35)(24,40,25,41,30,46,42,44)

Polycyclic presentation:

2^9|b^2=i,c^2=fhi,d^2=gi,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=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, 6733824);