#6683253

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

1913 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)(9,10,15,19)(11,21,23,16)(12,25,24,36)(13,20,26,17)(14,18,30,35)(22,38,39,32)(27,34,29,40)(28,37,31,33);(1,3,4,8)(2,5,6,7)(9,11,15,23)(10,16,19,21)(12,26,24,13)(14,29,30,27)(17,25,20,36)(18,34,35,40)(22,31,39,28)(32,33,38,37);(9,12,26,11,15,24,13,23)(10,16,20,36,19,21,17,25)(14,27,31,22,30,29,28,39)(18,32,37,40,35,38,33,34);(2,6)(5,7)(9,13,15,26)(10,17,19,20)(11,12,23,24)(14,28,30,31)(16,25,21,36)(18,33,35,37)(22,27,39,29)(32,34,38,40);(3,8)(5,7)(9,14)(10,18)(11,22)(12,27)(13,28)(15,30)(16,32)(17,33)(19,35)(20,37)(21,38)(23,39)(24,29)(25,34)(26,31)(36,40)

Polycyclic presentation:

2^9|a^2=i,b^2=f,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,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, 6590449);