#6467853

order	512
nilpotency class	3
isoclinism	isoclinic class of #6443052
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1764 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)(9,10,16,19)(11,14,22,15)(12,17,25,23)(13,18,32,28)(20,26,36,27)(21,30,38,31)(24,33,29,34)(35,39,37,40);(9,11,14,19,16,22,15,10)(12,23,26,36,25,17,27,20)(13,28,30,38,32,18,31,21)(24,37,39,33,29,35,40,34);(2,5)(6,8)(9,12,15,27,16,25,14,26)(10,17,11,20,19,23,22,36)(13,29,31,39,32,24,30,40)(18,34,21,37,28,33,38,35);(9,13,15,31,16,32,14,30)(10,18,11,21,19,28,22,38)(12,24,27,40,25,29,26,39)(17,33,20,35,23,34,36,37);(1,3,4,7)(2,6,5,8)(12,25)(17,23)(20,36)(24,29)(26,27)(33,34)(35,37)(39,40)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=h,d^2=h,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,h^b=hi,d^c=di,e^c=ei

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, 6375049);