#6811129

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806815
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

1761 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)(9,10,14,19)(11,22,25,17)(12,27,28,38)(13,20,29,39)(15,31,16,21)(18,32,34,24)(23,40,30,37)(26,35,33,36);(1,2)(3,6)(4,8)(5,7)(9,11)(10,17)(12,13)(14,25)(15,26)(16,33)(18,37)(19,22)(20,38)(21,35)(23,24)(27,39)(28,29)(30,32)(31,36)(34,40);(9,12,16,24,14,28,15,32)(10,18,21,38,19,34,31,27)(11,23,26,13,25,30,33,29)(17,20,36,37,22,39,35,40);(1,3,5,4)(2,6,7,8)(10,19)(17,22)(18,34)(20,39)(21,31)(27,38)(35,36)(37,40);(1,4,5,3)(2,6,7,8)(9,13,15,23,14,29,16,30)(10,20,31,40,19,39,21,37)(11,24,33,12,25,32,26,28)(17,34,35,38,22,18,36,27)

Polycyclic presentation:

2^9|a^2=f,c^2=h,d^2=fi,e^2=fh,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,g^c=gi

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, 6718325);