#497435

order	512
nilpotency class	4
isoclinism	isoclinic class of #33830
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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×#508	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1022 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)(5,6,7,8)(9,10)(11,19)(12,18)(13,27)(15,16)(17,22)(20,32)(21,33)(23,36)(25,26)(28,30)(31,35)(34,40)(37,39);(9,11,16,21)(10,17,15,30)(12,20,26,34)(13,28,27,22)(14,19,29,33)(18,31,25,39)(23,37,36,35)(24,32,38,40);(2,4)(9,12,16,26)(10,18,15,25)(11,20,21,34)(13,23,27,36)(14,24,29,38)(17,31,30,39)(19,32,33,40)(22,35,28,37);(9,13)(10,14)(11,21)(12,23)(15,29)(16,27)(18,24)(19,28)(20,34)(22,33)(25,38)(26,36)(32,37)(35,40)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,h^a=hi,d^b=de,e^b=eh,h^b=hi,e^d=eh,h^d=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, 404631);