#494049

order	512
nilpotency class	4
isoclinism	isoclinic class of #481284
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

978 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)(9,10,13,18,14,20,28,19)(11,15,23,32,21,30,36,33)(12,24,26,17,27,16,25,34)(22,37,39,31,35,29,38,40);(2,5)(6,8)(9,11,14,21)(10,15,20,30)(12,22,27,35)(13,23,28,36)(16,29,24,37)(17,31,34,40)(18,32,19,33)(25,38,26,39);(10,16)(12,25)(13,28)(15,29)(17,18)(19,34)(20,24)(22,38)(23,36)(26,27)(30,37)(31,32)(33,40)(35,39);(1,3,4,7)(2,6,5,8)(11,21)(15,30)(22,35)(23,36)(29,37)(31,40)(32,33)(38,39)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401245);