#6682656

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682648
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2152 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)(3,7)(4,6)(5,8)(9,10)(11,21)(12,26)(13,20)(14,19)(15,18)(16,25)(17,30)(22,38)(23,33)(24,31)(27,32)(28,36)(29,35)(34,40)(37,39);(1,3)(2,5)(4,8)(6,7)(9,11,14,25)(10,16,19,21)(12,27,29,39)(13,31,15,23)(17,34,36,38)(18,24,20,33)(22,28,40,30)(26,37,35,32);(9,12,13,28,14,29,15,30)(10,17,18,35,19,36,20,26)(11,22,31,27,25,40,23,39)(16,32,24,38,21,37,33,34);(2,6)(5,7)(9,13,14,15)(10,18,19,20)(11,23,25,31)(12,28,29,30)(16,33,21,24)(17,35,36,26)(22,39,40,27)(32,34,37,38);(3,8)(5,7)(9,10)(11,24)(12,17)(13,18)(14,19)(15,20)(16,23)(21,31)(22,38)(25,33)(26,30)(27,37)(28,35)(29,36)(32,39)(34,40)

Polycyclic presentation:

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

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, 6589852);