#504260

order	512
nilpotency class	4
isoclinism	isoclinic class of #500733
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

534 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=f,c^2=f,d^2=ei,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gh,h^b=hi,d^c=di,g^c=gh,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411456);