#6577912

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573652
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

1873 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)(3,7)(4,8)(5,6)(9,10)(11,16)(12,26)(13,18)(14,20)(15,21)(17,28)(19,33)(22,35)(23,24)(25,31)(27,38)(29,36)(30,40)(32,37)(34,39);(2,6)(7,8)(9,11,15,13)(10,16,21,18)(12,27,29,39)(14,24,33,25)(17,35,37,40)(19,31,20,23)(22,32,30,28)(26,38,36,34);(9,12,15,29)(10,17,21,37)(11,22,13,30)(14,32,33,28)(16,34,18,38)(19,36,20,26)(23,35,31,40)(24,27,25,39);(1,3,5,4)(2,7,6,8)(9,13,15,11)(10,18,21,16)(12,27,29,39)(14,25,33,24)(17,35,37,40)(19,23,20,31)(22,32,30,28)(26,38,36,34);(1,4,5,3)(2,8,6,7)(9,10)(11,23)(12,17)(13,31)(14,19)(15,21)(16,24)(18,25)(20,33)(22,35)(26,28)(27,34)(29,37)(30,40)(32,36)(38,39)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=fi,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,h^b=hi,d^c=dh,g^c=gi,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6485108);