#498329

order	512
nilpotency class	4
isoclinism	isoclinic class of #33830
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

644 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405525);