#6454807

order	512
nilpotency class	3
isoclinism	isoclinic class of #24468
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×#120
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3005 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dhi,h^b=hi,h^c=hi,h^d=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, 6362003);