Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #17082 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#17296 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C24 |
| commutator, Frattini, cosocle | C24 | C25 |
| prime radical | C26 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3502 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(4,7)(5,9)(11,12,14,15)(13,17,18,16)(19,20,22,24)(21,23,25,26)(27,28,31,34)(29,30,33,32);(5,9)(8,10)(12,15)(16,17)(21,25)(23,26)(27,29,31,33)(28,32,34,30);(3,4,6,7)(5,8,9,10)(11,12)(13,17)(14,15)(16,18)(19,21)(20,23)(22,25)(24,26)(27,29)(28,32)(30,34)(31,33);(3,5,6,9)(4,8,7,10)(11,13,14,18)(12,16,15,17)(19,20)(21,23)(22,24)(25,26)(27,30,31,32)(28,33,34,29)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 752362);