#502504

order	512
nilpotency class	4
isoclinism	isoclinic class of #500839
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
prime radical	C₂×#2218	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

713 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=efhi,d^2=ehi,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,d^b=di,d^c=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, 409700);