#6351211

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351101
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 ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2758 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)(3,12)(4,9)(5,10)(6,15)(7,11)(8,13)(14,16)(17,18,23,21)(19,27,30,25)(20,24,31,29)(22,32,28,26);(1,3,7,4)(2,8,11,16)(5,13,6,14)(9,15,12,10)(17,19)(18,24)(20,28)(21,29)(22,31)(23,30)(25,26)(27,32);(1,4,7,3)(2,9,11,12)(5,13,6,14)(8,15,16,10)(17,20,23,31)(18,25,21,27)(19,28,30,22)(24,26,29,32);(1,5,7,6)(2,10,11,15)(3,13,4,14)(8,9,16,12)(17,21,23,18)(19,29,30,24)(20,25,31,27)(22,32,28,26);(2,11)(8,16)(9,12)(10,15)(18,21)(24,29)(25,27)(26,32)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6258407);