Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | C2×#29458 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1808 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);(3,4)(5,9)(6,8)(7,10)(19,20,25,28,24,30,23,29)(21,26,31,34,22,33,32,27);(3,5,6,10)(4,7,8,9)(11,12)(13,16)(14,15)(17,18)(19,21)(20,26)(22,24)(23,32)(25,31)(27,29)(28,34)(30,33);(12,15)(16,18)(19,22,24,21)(20,27,30,34)(23,32,25,31)(26,28,33,29);(4,8)(7,9)(11,13,14,17)(12,16,15,18)(19,23,24,25)(20,28,30,29)(21,31,22,32)(26,27,33,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6368813);