#6500161

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500157
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 ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3185 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,e^b=eh,f^b=fi,d^c=dh,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, 6407357);