#6809308

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806827
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#61055	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2161 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)(11,16)(12,24)(13,27)(14,18)(15,19)(17,30)(20,36)(21,35)(22,25)(23,32)(26,29)(28,40)(31,38)(33,37)(34,39);(2,5)(3,7)(9,11)(10,16)(12,25)(13,28)(14,21)(15,23)(17,33)(18,35)(19,32)(20,29)(22,24)(26,36)(27,40)(30,37)(31,34)(38,39);(9,12,14,10,15,18,24,19)(11,16,22,25,23,32,21,35)(13,26,29,17,30,34,39,27)(20,31,28,33,38,36,37,40);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(9,13,24,39,15,30,14,29)(10,17,12,26,19,27,18,34)(11,20,21,37,23,38,22,28)(16,31,35,40,32,36,25,33)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6716504);