#512511

order	512
nilpotency class	4
isoclinism	isoclinic class of #511951
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

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

510 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(18,24)(19,29)(20,21)(22,33)(25,36)(27,38)(28,40)(30,31)(32,44)(34,35)(37,41)(42,47)(43,45)(46,48);(2,5)(6,8)(17,18,21,27,22,28,34,36,23,25,35,40,33,38,20,24)(19,26,29,41,31,42,43,46,32,39,44,48,45,47,30,37);(9,11,12,16)(10,13,14,15)(18,25)(24,36)(26,39)(27,40)(28,38)(37,46)(41,48)(42,47);(1,3,4,7)(2,6,5,8)(17,19,20,30,33,45,35,44,23,32,34,43,22,31,21,29)(18,26,24,37,38,47,40,48,25,39,36,46,28,42,27,41)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=e,e^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=ghi,h^a=hi,c^b=ci

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, 419707);