#6452227

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437931
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	#60444	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3281 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)(3,8)(4,6)(5,7)(9,10,15,19)(11,20,22,32)(12,17,26,36)(13,27,14,18)(16,28,31,23)(21,33,37,40)(24,39,38,30)(25,35,29,34);(9,11,13,23,15,22,14,28)(10,16,18,20,19,31,27,32)(12,24,25,21,26,38,29,37)(17,33,34,39,36,40,35,30);(1,3,5,4)(2,6,7,8)(9,12)(10,17)(11,21)(13,25)(14,29)(15,26)(16,30)(18,34)(19,36)(20,33)(22,37)(23,38)(24,28)(27,35)(31,39)(32,40);(1,4,5,3)(2,6,7,8)(9,12)(10,17)(11,21)(13,25)(14,29)(15,26)(16,30)(18,34)(19,36)(20,33)(22,37)(23,38)(24,28)(27,35)(31,39)(32,40);(11,22)(16,31)(20,32)(21,37)(23,28)(24,38)(30,39)(33,40)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=g,d^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^b=ei,h^b=hi,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, 6359423);