#502102

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

825 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)(9,10,14,19,12,26,27,40,13,18,28,35,15,17,30,29)(11,20,24,38,22,33,36,21,23,16,31,34,25,37,32,39);(2,4)(5,6)(7,8)(9,11,14,24,12,22,27,36,13,23,28,31,15,25,30,32)(10,16,19,34,26,37,40,39,18,20,35,38,17,33,29,21);(6,8)(9,10)(11,21)(12,17)(13,18)(14,29)(15,26)(16,32)(19,30)(20,36)(22,38)(23,39)(24,33)(25,34)(27,35)(28,40)(31,37);(2,4)(9,12,13,15)(10,17,18,26)(11,22,23,25)(14,27,28,30)(16,33,20,37)(19,29,35,40)(21,38,39,34)(24,36,31,32)

Polycyclic presentation:

2^9|a^2=f,b^2=f,d^2=i,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^c=fh,g^c=gh,h^c=hi

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, 409298);