#6836919

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1625 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gh,d^2=f,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,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, 6744115);