#507522

order	512
nilpotency class	4
isoclinism	isoclinic class of #501064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

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

468 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=egh,b^2=eg,c^2=ef,d^2=e,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=cg,d^b=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414718);