#502490

order	512
nilpotency class	4
isoclinism	isoclinic class of #500829
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	#13304	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

729 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)(11,21,12,26)(13,29,31,33)(15,32,30,20)(16,27,18,24)(19,22,35,23)(25,40,28,37)(34,39,36,38);(1,2)(3,4)(5,7,8,6)(9,11,22,38,15,25,13,24,14,12,23,39,30,28,31,27)(10,16,33,40,20,34,19,21,17,18,29,37,32,36,35,26);(2,4)(6,7)(10,17)(16,18)(19,35)(20,32)(21,26)(29,33)(34,36)(37,40);(9,12,22,39,15,28,13,27,14,11,23,38,30,25,31,24)(10,18,33,37,20,36,19,26,17,16,29,40,32,34,35,21)

Polycyclic presentation:

2^9|a^2=f,b^2=efh,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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409686);