#494135

order	512
nilpotency class	3
isoclinism	isoclinic class of #188764
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

560 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)(9,10,14,21,13,16,28,33,15,20,31,18,29,19,12,27)(11,22,26,36,25,34,24,39,23,17,30,35,38,32,37,40);(2,5)(6,8)(10,16,20,19)(11,23)(17,34,22,32)(18,27,21,33)(24,37)(25,38)(26,30)(35,39,36,40);(9,11)(10,17)(12,24)(13,25)(14,30)(15,23)(16,32)(18,35)(19,34)(20,22)(21,36)(26,31)(27,40)(28,37)(29,38)(33,39);(1,3,4,7)(2,6,5,8)(9,12,13,14,15,28,29,31)(10,18,19,33,20,21,16,27)(11,24,25,30,23,37,38,26)(17,35,34,39,22,36,32,40)

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,c^b=ci,g^c=gi,g^d=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401331);