#6685511

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

1766 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)(3,6)(4,5)(7,8)(9,10,15,14)(11,21,25,17)(12,27,31,36)(13,20,16,19)(18,29,30,32)(22,35,38,40)(23,34,26,24)(28,37,39,33);(2,5)(3,7)(9,11)(10,17)(12,28)(13,26)(14,21)(15,25)(16,23)(18,35)(19,24)(20,34)(22,29)(27,33)(30,40)(31,39)(32,38)(36,37);(9,12,13,29,15,31,16,32)(10,18,19,36,14,30,20,27)(11,22,26,39,25,38,23,28)(17,33,24,35,21,37,34,40);(2,5)(6,8)(9,13,15,16)(10,19,14,20)(11,23,25,26)(12,29,31,32)(17,34,21,24)(18,36,30,27)(22,28,38,39)(33,40,37,35);(1,3,4,7)(2,6,5,8)(9,14,15,10)(11,24,25,34)(12,30,31,18)(13,20,16,19)(17,26,21,23)(22,35,38,40)(27,32,36,29)(28,33,39,37)

Polycyclic presentation:

2^9|a^2=i,c^2=gi,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, 6592707);