#501317

order	512
nilpotency class	4
isoclinism	isoclinic class of #501052
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

730 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)(3,7)(4,6)(5,8)(9,10,15,28,26,37,27,17,12,18,13,19,14,20,30,36)(11,21,29,32,25,34,38,40,23,16,24,31,22,33,39,35);(1,3,4,8)(2,5,6,7)(9,11,12,23)(10,16,18,21)(13,24,15,29)(14,22,26,25)(17,35,36,40)(19,32,28,31)(20,34,37,33)(27,38,30,39);(10,17)(11,22)(13,27)(14,26)(15,30)(16,31)(18,36)(19,20)(21,32)(23,25)(24,29)(28,37)(33,40)(34,35);(2,6)(5,7)(10,18)(16,21)(17,36)(19,28)(20,37)(31,32)(33,34)(35,40)

Polycyclic presentation:

2^9|a^2=fi,b^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,f^c=fh,g^c=gi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408513);