#6817231

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

Polycyclic presentation:

2^9|c^2=fgh,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,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, 6724427);