#501678

order	512
nilpotency class	4
isoclinism	isoclinic class of #500834
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

651 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)(12,26)(13,29)(14,17)(15,32)(16,24)(18,28)(19,22)(20,30)(23,37)(25,40)(27,33)(31,36)(34,39)(35,38);(5,6)(7,8)(9,11,22,38,15,25,31,39,14,23,27,28,30,12,13,24)(10,16,29,26,20,18,33,37,17,34,36,40,32,35,19,21);(2,4)(6,8)(10,17)(11,23)(12,25)(19,33)(20,32)(24,39)(28,38)(29,36);(9,12,27,39,15,11,13,28,14,25,22,24,30,23,31,38)(10,18,36,21,20,34,19,26,17,35,29,37,32,16,33,40)

Polycyclic presentation:

2^9|a^2=fi,b^2=eh,d^2=ehi,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^c=di

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, 408874);