#6617476

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573671
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3047 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:

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

Polycyclic presentation:

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

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, 6524672);