#494130

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

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,c^b=ci,d^b=di,g^b=gi,d^c=di,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, 401326);