Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
| direct product | C4×#2817 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C23 | C24×C4 |
| Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2614 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,3,4);(9,10)(11,13)(12,14)(15,16)(17,18,22,25)(19,26,20,23)(21,24,29,32)(27,31,28,30);(5,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,20)(18,23,25,26)(21,28,29,27)(24,31,32,30);(6,8)(10,14)(13,16)(17,20,22,19)(18,23,25,26)(21,27,29,28)(24,31,32,30);(11,15)(13,16)(17,21,22,29)(18,24,25,32)(19,27,20,28)(23,30,26,31)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 451529);