#503262

order	512
nilpotency class	4
isoclinism	isoclinic class of #500839
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#509	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1158 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=eh,c^2=fi,d^2=eh,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,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, 410458);