#503148

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×#3285	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

1160 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)(6,8)(9,10,16,21)(11,17,24,37)(12,25,23,38)(13,19,15,32)(14,30,31,40)(18,33,36,22)(20,26,34,27)(28,39,29,35);(1,2,4,3)(9,11)(10,17)(12,26)(13,22)(14,29)(15,33)(16,24)(18,19)(20,38)(21,37)(23,27)(25,34)(28,31)(30,35)(32,36)(39,40);(5,6,7,8)(9,12,27,24,15,28,31,33,16,23,26,11,13,29,14,22)(10,18,30,35,19,17,34,38,21,36,40,39,32,37,20,25);(1,3,4,2)(9,13,16,15)(10,19,21,32)(11,22,24,33)(12,28,23,29)(14,26,31,27)(17,18,37,36)(20,30,34,40)(25,39,38,35)

Polycyclic presentation:

2^9|a^2=i,b^2=e,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, 410344);