#6491426

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470058
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 ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2931 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:

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

Polycyclic presentation:

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

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, 6398622);