#6349424

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349370
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	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2665 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 6256620);