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 | |
4864 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(9,10,13,11)(12,15,16,14)(17,18)(19,24)(20,28)(21,26)(22,23)(25,30)(27,29)(31,32);(1,2,5,4)(3,7,8,6)(10,11)(14,15)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,30)(29,32);(2,4)(6,7)(9,11,13,10)(12,15,16,14)(18,23)(21,24)(25,28)(29,31);(1,3)(2,6)(4,7)(5,8)(9,12)(10,14)(11,15)(13,16)(17,19,22,26)(18,24,23,21)(20,27,30,32)(25,31,28,29);(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,29,24,31);(1,4,5,2)(3,7,8,6)(17,21,22,24)(18,26,23,19)(20,29,30,31)(25,32,28,27)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=g,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8102987);