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×#47537 |
| 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)(20,25)(21,28)(23,31)(30,33);(3,4)(5,9)(6,8)(7,10)(19,20,24,25)(21,22,28,26)(23,32,31,27)(29,33,34,30);(3,4,6,8)(5,7,10,9)(11,12)(13,15)(14,16)(17,18)(19,21,24,28)(20,26,25,22)(23,29,31,34)(27,33,32,30);(3,4)(5,7)(6,8)(9,10)(19,22,24,26)(20,21,25,28)(23,30,31,33)(27,29,32,34);(3,5,6,10)(4,7,8,9)(11,13,14,17)(12,15,16,18)(19,23,24,31)(20,27,25,32)(21,29,28,34)(22,30,26,33)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7608101);