#6682634

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682628
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

2151 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)(3,7)(4,6)(5,8)(9,10)(11,16)(12,25)(13,18)(14,19)(15,21)(17,24)(20,28)(22,31)(23,32)(26,35)(27,34)(29,36)(30,37)(33,38)(39,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,13)(14,26)(15,24)(17,21)(18,25)(19,35)(20,32)(22,29)(23,28)(27,38)(30,39)(31,36)(33,34)(37,40);(9,12,13,24,15,23,28,11)(10,17,20,25,21,16,18,32)(14,26,27,38,30,39,29,22)(19,33,36,35,37,31,34,40);(2,6)(5,7)(9,13,15,28)(10,18,21,20)(11,12,24,23)(14,27,30,29)(16,25,17,32)(19,34,37,36)(22,26,38,39)(31,35,33,40);(3,8)(5,7)(9,14)(10,19)(11,22)(12,26)(13,27)(15,30)(16,31)(17,33)(18,34)(20,36)(21,37)(23,39)(24,38)(25,35)(28,29)(32,40)

Polycyclic presentation:

2^9|c^2=g,d^2=i,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,h^c=hi,h^e=hi

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, 6589830);