#494125

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 ◁ C₂×#405 ◁ 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)(3,5)(4,6)(7,8)(9,10,12,18,13,19,24,33,14,15,23,27,25,17,30,32)(11,16,21,29,22,28,36,38,20,26,34,37,35,31,40,39);(1,3,4,7)(2,5,6,8)(10,15)(12,23)(16,26)(17,19)(21,34)(24,30)(28,31)(36,40);(2,6)(5,8)(9,11)(10,16)(12,21)(13,22)(14,20)(15,26)(17,31)(18,29)(19,28)(23,34)(24,36)(25,35)(27,37)(30,40)(32,39)(33,38);(10,17,15,19)(11,20)(12,23)(16,28,26,31)(18,33,27,32)(22,35)(24,30)(29,39,37,38)

Polycyclic presentation:

2^9|a^2=g,b^2=f,d^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^b=gi,d^c=di,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, 401321);