Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#47538 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4914 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)(5,7)(6,8)(9,10)(11,12,14,16)(13,15,17,18)(19,20,25,23)(21,28,26,22)(24,31,33,27)(29,30,32,34);(3,4)(5,9)(6,8)(7,10)(19,21,25,26)(20,22,23,28)(24,32,33,29)(27,34,31,30);(3,4,6,8)(5,7,10,9)(11,12)(13,15)(14,16)(17,18)(19,22,25,28)(20,26,23,21)(24,30,33,34)(27,32,31,29);(3,4)(5,7)(6,8)(9,10)(19,23,25,20)(21,22,26,28)(24,31,33,27)(29,30,32,34);(3,5,6,10)(4,7,8,9)(11,13,14,17)(12,15,16,18)(19,24,25,33)(20,27,23,31)(21,29,26,32)(22,30,28,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7608117);