#6841335

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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 ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1244 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)(11,15)(12,16)(13,14)(17,18,21,25)(19,27,30,23)(20,24,33,39)(22,26,34,40)(28,41,43,35)(29,38,32,36)(31,37,44,47)(42,48,45,46);(1,2,4,5)(3,6,7,8)(19,28)(20,31)(22,34)(23,35)(24,37)(26,40)(27,41)(29,32)(30,43)(33,44)(36,38)(39,47);(10,14)(15,16)(17,19,22,28,21,30,34,43)(18,23,26,35,25,27,40,41)(20,32,31,45,33,29,44,42)(24,38,37,48,39,36,47,46);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,15,14,16)(20,33)(24,39)(29,32)(31,44)(36,38)(37,47)(42,45)(46,48);(2,5)(6,8)(9,12,13,11)(10,16,14,15)(17,20,34,44,21,33,22,31)(18,24,40,47,25,39,26,37)(19,29,43,45,30,32,28,42)(23,36,41,48,27,38,35,46)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=h,d^2=fgi,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,e^d=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, 6748531);