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×#37393 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5706 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);(3,4,7,5)(6,9,10,8)(11,12,14,16)(13,17,18,15)(19,20)(21,22)(23,31)(24,25)(26,29)(27,32)(28,34)(30,33);(4,5)(8,9)(11,13,14,18)(12,15,16,17)(20,25)(22,29)(27,31)(30,34);(3,5,7,4)(6,9,10,8)(11,13)(12,15)(14,18)(16,17)(19,21,24,26)(20,22,25,29)(23,28,32,33)(27,30,31,34);(3,6)(4,8)(5,9)(7,10)(11,12)(13,15)(14,16)(17,18)(19,22)(20,26)(21,25)(23,30)(24,29)(27,33)(28,31)(32,34);(19,23,24,32)(20,27,25,31)(21,28,26,33)(22,30,29,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7541975);