#6595665

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573671
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3127 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)(11,21)(12,25)(13,20)(14,19)(15,18)(16,23)(17,26)(22,30)(24,33)(27,34)(28,31)(29,32)(35,37)(36,39)(38,40);(1,2,4,5)(3,6,7,8)(9,11,15,24)(10,16,18,31)(12,22,27,38)(13,28,14,23)(17,30,32,40)(19,33,20,21)(25,39,34,35)(26,37,29,36);(2,5)(3,7)(9,12)(10,17)(11,22)(13,26)(14,29)(15,27)(16,30)(18,32)(19,25)(20,34)(21,35)(23,36)(24,38)(28,37)(31,40)(33,39);(1,3)(2,6)(4,7)(5,8)(10,18)(16,31)(17,32)(19,20)(21,33)(25,34)(30,40)(35,39);(2,5)(3,7)(10,18)(16,31)(17,32)(19,20)(21,33)(25,34)(30,40)(35,39)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6502861);