Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#37056 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4996 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);(4,8)(6,9)(12,15)(16,18)(19,20,25,28)(21,23,22,26)(24,32,33,27)(29,34,30,31);(4,8)(5,10)(19,21,25,22)(20,26,28,23)(24,29,33,30)(27,34,32,31);(3,4,7,8)(5,9,10,6)(11,12)(13,16)(14,15)(17,18)(19,22,25,21)(20,26,28,23)(24,30,33,29)(27,34,32,31);(3,5)(4,9)(6,8)(7,10)(19,23,25,26)(20,21,28,22)(24,31,33,34)(27,29,32,30);(3,6,7,9)(4,5,8,10)(11,13,14,17)(12,16,15,18)(19,24,25,33)(20,27,28,32)(21,29,22,30)(23,31,26,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7541553);