#6809298

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806815
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	#61056	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2181 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)(13,17)(14,18)(15,19)(20,30)(21,35)(22,25)(23,31)(26,38)(27,32)(28,33)(29,34)(36,40)(37,39);(2,5)(3,7)(9,11)(10,16)(12,25)(13,27)(14,21)(15,23)(17,32)(18,35)(19,31)(20,28)(22,24)(26,39)(29,36)(30,33)(34,40)(37,38);(9,12,14,10,15,18,24,19)(11,16,22,25,23,31,21,35)(13,26,28,17,29,33,38,34)(20,30,27,32,37,39,36,40);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(9,13,24,38,15,29,14,28)(10,17,12,26,19,34,18,33)(11,20,21,36,23,37,22,27)(16,30,35,40,31,39,25,32)

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,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi

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, 6716494);