#501230

order	512
nilpotency class	4
isoclinism	isoclinic class of #499724
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

1183 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)(9,10,15,21)(11,20,14,29)(12,17,26,35)(13,18,28,31)(16,27,19,23)(22,34,25,39)(24,32,38,40)(30,37,33,36);(5,6)(7,8)(9,11)(10,16)(12,22)(14,15)(17,30)(18,20)(19,21)(23,27)(25,26)(29,31)(32,34)(33,35)(36,37)(39,40);(2,4)(6,8)(9,12,15,26)(10,17,21,35)(11,22,14,25)(13,24,28,38)(16,30,19,33)(18,32,31,40)(20,34,29,39)(23,36,27,37);(10,18)(11,14)(13,27)(16,29)(17,32)(19,20)(21,31)(22,25)(23,28)(24,37)(30,39)(33,34)(35,40)(36,38)

Polycyclic presentation:

2^9|a^2=fi,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,d^b=di,e^b=eh,h^b=hi,e^d=eh,h^d=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, 408426);