#6817232

order	512
nilpotency class	3
isoclinism	isoclinic class of #6814811
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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

1908 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:

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

Polycyclic presentation:

2^9|c^2=fgh,d^2=f,e^2=fhi,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,d^c=di

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, 6724428);