Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #15257 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#16161 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C24 |
| commutator, Frattini, cosocle | C24 | C25 |
| prime radical | C26 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4226 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(4,7)(8,10)(11,12,15,13)(14,17,18,16)(19,20)(21,24)(22,23)(25,26)(28,31)(32,33);(4,7)(8,10)(12,13)(14,18)(20,22)(24,25)(27,28,30,31)(29,33,34,32);(3,4,6,7)(5,8,9,10)(11,13,15,12)(14,17,18,16)(19,21,23,26)(20,24,22,25)(27,28)(29,32)(30,31)(33,34);(3,5,6,9)(4,8,7,10)(11,14,15,18)(12,16,13,17)(19,22,23,20)(21,25,26,24)(27,29)(28,32)(30,34)(31,33)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 517457);