#6574111

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574107
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	D₄×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2468 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|b^2=fi,d^2=i,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,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, 6481307);