#501672

order	512
nilpotency class	4
isoclinism	isoclinic class of #500829
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

664 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)(13,29)(14,17)(15,32)(16,23)(18,27)(19,26)(20,30)(22,35)(24,37)(28,38)(31,33)(34,40)(36,39);(5,6)(7,8)(9,11,22,39,15,12,13,23,14,24,26,27,30,28,31,40)(10,16,33,25,20,36,19,21,17,34,29,38,32,18,35,37);(2,4)(6,8)(10,17)(16,34)(18,36)(19,35)(20,32)(21,37)(25,38)(29,33);(9,12,26,40,15,24,31,39,14,28,22,23,30,11,13,27)(10,18,29,21,20,16,35,38,17,36,33,37,32,34,19,25)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408868);