#6456444

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438886
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60703	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1770 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6363640);