#6466966

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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1464 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)(9,10,13,15)(11,14,12,16)(17,18,21,25)(19,27,30,23)(20,24,33,38)(22,26,32,37)(28,39,42,34)(29,40,43,35)(31,36,44,46)(41,47,48,45);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(19,28)(20,31)(22,32)(23,34)(24,36)(26,37)(27,39)(29,43)(30,42)(33,44)(35,40)(38,46);(9,12,13,11)(10,14,15,16)(17,19,22,28,21,30,32,42)(18,23,26,34,25,27,37,39)(20,29,31,41,33,43,44,48)(24,35,36,45,38,40,46,47);(17,20,32,44,21,33,22,31)(18,24,37,46,25,38,26,36)(19,29,42,48,30,43,28,41)(23,35,39,47,27,40,34,45);(1,3,4,7)(2,6,5,8)(19,30)(23,27)(28,42)(29,43)(34,39)(35,40)(41,48)(45,47)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=ghi,d^2=hi,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,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, 6374162);