#498311

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

664 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=i,d^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci,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, 405507);