Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4830 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2)(3,4)(5,7)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18,23,21)(19,27,29,24)(20,30,31,25)(22,32,28,26);(1,3,6,5)(2,7,8,4)(9,10,12,14)(11,13,16,15)(17,19,23,29)(18,24,21,27)(20,28,31,22)(25,26,30,32);(9,10)(11,13)(12,14)(15,16)(17,20,23,31)(18,25,21,30)(19,22,29,28)(24,32,27,26);(1,4,6,7)(2,5,8,3)(9,11)(10,13)(12,16)(14,15)(17,20)(18,25)(19,28)(21,30)(22,29)(23,31)(24,26)(27,32);(1,5,6,3)(2,7,8,4)(17,21,23,18)(19,24,29,27)(20,30,31,25)(22,32,28,26);(1,2)(3,7)(4,5)(6,8)(17,22,23,28)(18,26,21,32)(19,20,29,31)(24,30,27,25)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=h,d^2=gi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10075650);