#509059

order	512
nilpotency class	4
isoclinism	isoclinic class of #508298
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

415 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416255);