#6502944

order	512
nilpotency class	3
isoclinism	isoclinic class of #6502937
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 ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1930 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6410140);