#494131

order	512
nilpotency class	3
isoclinism	isoclinic class of #188761
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

574 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

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,d^a=dh,e^a=ei,f^a=fi,g^c=gi,g^d=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401327);