#501218

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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 ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1122 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408414);