#494121

order	512
nilpotency class	3
isoclinism	isoclinic class of #20671
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#432
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

584 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,d^2=e,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,f^a=fi,c^b=ci,g^b=gi,d^c=di,g^c=gi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401317);