#6347206

order	512
nilpotency class	3
isoclinism	isoclinic class of #6347203
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 ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

3135 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:

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

Polycyclic presentation:

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

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, 6254402);