#501909

order	512
nilpotency class	4
isoclinism	isoclinic class of #501810
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#481
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₄	SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#435 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

478 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)(11,20)(14,25)(18,35)(21,37)(22,30)(24,28)(26,40)(36,39);(2,5)(6,8)(9,10,12,17,13,19,15,29)(11,21,30,39,20,37,22,36)(14,26,24,35,25,40,28,18)(16,31,33,23,34,27,32,38);(1,3,4,7)(2,6,5,8)(9,11,13,20)(10,16,19,34)(12,22,15,30)(14,27,25,31)(17,32,29,33)(18,36,35,39)(21,26,37,40)(23,28,38,24);(2,5)(6,8)(9,12,13,15)(10,17,19,29)(11,22,20,30)(14,24,25,28)(16,32,34,33)(18,26,35,40)(21,36,37,39)(23,31,38,27)

Polycyclic presentation:

2^9|b^2=fh,c^2=ei,d^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,g^b=gh,h^b=hi,g^c=ghi,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, 409105);