Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#22140 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C24 | C23×C4 |
| prime radical | C2×#392 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2496 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 30 points:
(1,2);(3,4)(5,6)(8,11)(9,13)(15,16,20,24,19,23,18,22)(17,21,25,28,26,29,27,30);(7,8,10,11)(9,12,13,14)(15,16,18,22,19,23,20,24)(17,21,27,30,26,29,25,28);(4,6)(8,11)(12,14)(15,17)(16,21)(18,27)(19,26)(20,25)(22,30)(23,29)(24,28);(7,9,10,13)(8,12,11,14)(15,18,19,20)(16,22,23,24)(17,25,26,27)(21,28,29,30)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6283000);