#9137476

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4942 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=hi,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9044672);