#6432090

order	512
nilpotency class	3
isoclinism	isoclinic class of #6431723
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	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3276 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339286);