#6686985

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682638
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 ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1812 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=g,d^2=f,e^2=f,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,e^d=ei,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, 6594181);