#501959

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	C₂³	C₂²×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#446 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

417 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=fh,c^2=i,d^2=e,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, 409155);