#6467983

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1330 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6375179);