#503117

order	512
nilpotency class	4
isoclinism	isoclinic class of #501064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#2186	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1267 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:

(2,3)(9,10,13,18)(11,21,22,16)(12,24,27,39)(14,32,30,40)(15,33,31,20)(17,34,35,28)(19,36,25,26)(23,38,37,29);(1,2,4,3)(5,6)(7,8)(9,11,13,22)(10,16,18,21)(12,25,27,19)(14,23,30,37)(15,17,31,35)(20,34,33,28)(24,36,39,26)(29,40,38,32);(6,8)(9,12,26,16,15,29,30,34,13,27,36,21,31,38,14,28)(10,17,32,37,20,22,25,39,18,35,40,23,33,11,19,24);(1,3,4,2)(10,18)(11,22)(17,35)(19,25)(20,33)(23,37)(24,39)(32,40)

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=fh,d^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,d^b=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, 410313);