#503150

order	512
nilpotency class	4
isoclinism	isoclinic class of #499724
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×#3280	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1230 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

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, 410346);