#507809

order	512
nilpotency class	4
isoclinism	isoclinic class of #501064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

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

454 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)(3,7)(4,6)(5,8)(17,18,23,29,21,35,36,40,22,28,37,47,24,27,39,46)(19,30,32,45,33,43,38,26,20,25,34,42,31,41,48,44);(1,3,4,8)(2,5,6,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,20)(18,25,28,30)(21,33,24,31)(23,32,37,34)(26,40,44,46)(27,43,35,41)(29,42,47,45)(36,38,39,48);(10,13)(14,16)(17,20,22,19)(18,26,28,44)(21,33,24,31)(23,38,37,48)(25,40,30,46)(27,42,35,45)(29,43,47,41)(32,39,34,36);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,21,22,24)(18,27,28,35)(19,31,20,33)(23,36,37,39)(25,41,30,43)(26,42,44,45)(29,46,47,40)(32,48,34,38)

Polycyclic presentation:

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

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, 415005);