#501227

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
prime radical	C₂×#3285	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1088 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)(11,22)(12,18)(13,26)(14,29)(15,33)(16,21)(17,23)(19,27)(20,31)(24,39)(25,37)(28,36)(30,35)(32,38)(34,40);(5,6)(7,8)(9,11,16,24)(10,17,21,35)(12,13,15,28)(14,27,32,40)(18,19,33,34)(20,36,37,26)(22,38,39,29)(23,25,30,31);(2,4)(6,8)(9,12,16,15)(10,18,21,33)(11,13,24,28)(14,25,32,31)(17,19,35,34)(20,29,37,38)(22,26,39,36)(23,27,30,40);(9,13,16,28)(10,19,21,34)(11,12,24,15)(14,30,32,23)(17,18,35,33)(20,22,37,39)(25,40,31,27)(26,38,36,29)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^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, 408423);