#512492

order	512
nilpotency class	4
isoclinism	isoclinic class of #511951
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

463 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=eg,c^2=fhi,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,h^a=hi,c^b=ci,d^b=di

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, 419688);