#501064

order	512
nilpotency class	4
isoclinism	isoclinic class of #501064
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
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1196 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:

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

Polycyclic presentation:

2^9|a^2=f,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, 408260);