#6513322

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500006
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
prime radical	#62878	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3007 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)(11,16)(12,22,24,35)(13,26)(14,18)(15,19)(17,25,33,28)(20,37,23,40)(21,31)(27,34)(29,36)(30,38,32,39);(1,2)(3,6)(4,5)(7,8)(9,11,15,13)(10,16,19,26)(12,23,24,20)(14,27,29,21)(17,32,33,30)(18,34,36,31)(22,37,35,40)(25,38,28,39);(2,5)(3,7)(9,12)(10,17)(11,20)(13,23)(14,28)(15,24)(16,30)(18,35)(19,33)(21,39)(22,36)(25,29)(26,32)(27,38)(31,40)(34,37);(1,3,4,7)(2,6,5,8)(9,13,15,11)(10,16,19,26)(12,20,24,23)(14,27,29,21)(17,32,33,30)(18,31,36,34)(22,40,35,37)(25,38,28,39);(2,5)(6,8)(11,13)(16,26)(20,23)(21,27)(30,32)(31,34)(37,40)(38,39)

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=fi,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, 6420518);