#6685509

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

1810 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)(3,6)(4,5)(7,8)(9,10,15,14)(11,21,25,17)(12,26,30,36)(13,20,16,19)(18,28,29,31)(22,38,40,35)(23,24,32,34)(27,33,39,37);(2,5)(3,7)(9,11)(10,17)(12,27)(13,32)(14,21)(15,25)(16,23)(18,35)(19,34)(20,24)(22,31)(26,37)(28,40)(29,38)(30,39)(33,36);(9,12,16,31,15,30,13,28)(10,18,20,26,14,29,19,36)(11,22,23,39,25,40,32,27)(17,33,24,35,21,37,34,38);(2,5)(6,8)(9,13,15,16)(10,19,14,20)(11,23,25,32)(12,28,30,31)(17,24,21,34)(18,36,29,26)(22,39,40,27)(33,35,37,38);(1,3,4,7)(2,6,5,8)(9,14,15,10)(11,24,25,34)(12,29,30,18)(13,20,16,19)(17,23,21,32)(22,35,40,38)(26,31,36,28)(27,33,39,37)

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=i,e^2=f,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, 6592705);