Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#50750 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5392 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(11,12)(13,15)(14,18)(16,17)(20,21)(23,31)(25,27)(29,34);(3,4)(5,7)(6,9)(8,10)(11,13)(12,15)(14,16)(17,18)(19,20,24,21)(22,27,30,25)(23,26,31,28)(29,33,34,32);(4,5)(8,9)(12,17)(13,14)(19,21,24,20)(22,27,30,25)(23,28,31,26)(29,33,34,32);(3,5,7,4)(6,9,10,8)(11,14,16,13)(12,18,17,15)(19,22)(20,25)(21,27)(23,29)(24,30)(26,32)(28,33)(31,34);(3,6)(4,8)(5,9)(7,10)(11,15,16,18)(12,13,17,14)(19,23)(20,26)(21,28)(22,29)(24,31)(25,32)(27,33)(30,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7613516);