#6467854

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

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

Polycyclic presentation:

2^9|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,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, 6375050);