#494826

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂⁴
commutator, socle	C₂³	C₂³×C₈
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

842 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)(6,7)(9,10,14,19,15,20,29,37,13,18,22,31,28,36,32,39)(11,21,24,12,17,26,35,27,23,16,30,25,34,33,40,38);(5,6,8,7)(9,11,22,30,15,17,32,40,13,23,14,24,28,34,29,35)(10,16,31,12,20,33,39,27,18,21,19,25,36,26,37,38);(2,4)(5,7,8,6)(9,12,13,25)(10,17,18,34)(11,20,23,36)(14,26,22,33)(15,27,28,38)(16,32,21,29)(19,35,31,40)(24,37,30,39);(11,23)(12,25)(16,21)(17,34)(24,30)(26,33)(27,38)(35,40)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,d^b=di,d^c=di

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402022);