#6435415

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432881
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	#62856	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2920 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:

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

Polycyclic presentation:

2^9|a^2=fg,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,c^b=ch,f^b=fi,g^b=gi,f^c=fi,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, 6342611);