#497749

order	512
nilpotency class	3
isoclinism	isoclinic class of #188770
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³

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 ◁ #1752 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

456 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404945);