#501233

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	C₂²×D₄	D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1075 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)(11,21)(12,25)(13,28)(14,17)(15,32)(16,23)(18,26)(19,31)(20,22)(24,38)(27,36)(29,33)(30,37)(34,40)(35,39);(5,6)(7,8)(9,11,14,24)(10,16,17,33)(12,15,27,22)(13,26,30,40)(18,32,34,20)(19,36,35,25)(21,37,38,28)(23,39,29,31);(2,4)(6,8)(10,17)(16,33)(18,34)(19,35)(20,32)(21,38)(25,36)(28,37);(9,12,14,27)(10,18,17,34)(11,22,24,15)(13,29,30,23)(16,20,33,32)(19,21,35,38)(25,37,36,28)(26,39,40,31)

Polycyclic presentation:

2^9|a^2=fi,b^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, 408429);