#6544195

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533075
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

2844 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

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, 6451391);