#9159948

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

4801 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=ghi,d^2=ghi,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, 9067144);