#495321

order	512
nilpotency class	4
isoclinism	isoclinic class of #483839
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

514 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)(11,19,21,32)(12,23)(13,26)(15,27,28,40)(17,30)(18,29)(20,31,34,35)(22,37,36,33)(24,38)(25,39);(2,5)(6,8)(9,10)(11,20,21,34)(12,24)(13,18)(14,16)(15,22,28,36)(17,25)(19,31,32,35)(23,38)(26,29)(27,37,40,33)(30,39);(9,11,12,19,13,22,23,33,14,21,25,32,26,36,39,37)(10,15,17,27,18,20,30,35,16,28,24,40,29,34,38,31);(1,3,4,7)(2,6,5,8)(10,16)(15,28)(17,24)(18,29)(20,34)(27,40)(30,38)(31,35)

Polycyclic presentation:

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

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, 402517);