#6467857

order	512
nilpotency class	3
isoclinism	isoclinic class of #6443050
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

1748 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)(11,13)(12,26)(14,19)(15,21)(16,20)(17,24)(18,31)(22,28)(23,33)(25,34)(27,35)(29,38)(30,37)(32,36)(39,40);(9,11,15,24,16,25,14,23)(10,17,19,13,20,33,21,34)(12,27,30,22,31,39,29,36)(18,32,37,40,26,28,38,35);(2,5)(6,8)(9,12,14,29,16,31,15,30)(10,18,19,37,20,26,21,38)(11,22,24,39,25,36,23,27)(13,32,17,35,34,28,33,40);(9,13,15,33,16,34,14,17)(10,11,21,23,20,25,19,24)(12,28,30,35,31,32,29,40)(18,36,38,39,26,22,37,27);(1,3,4,7)(2,6,5,8)(9,14,16,15)(10,19,20,21)(11,23,25,24)(12,29,31,30)(13,33,34,17)(18,37,26,38)(22,27,36,39)(28,35,32,40)

Polycyclic presentation:

2^9|b^2=f,c^2=hi,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^d=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, 6375053);