Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
| direct product | C2×C4×#473 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #121 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C2×#121 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2652 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 38 points:
(1,2);(3,4,5,6);(7,8,13,18)(9,19,23,29)(10,24,26,32)(11,17,12,16)(14,22,30,21)(15,27,31,25)(20,34,36,28)(33,37,38,35);(7,9,12,22,13,23,11,21)(8,14,17,29,18,30,16,19)(10,20,27,37,26,36,25,35)(15,28,32,38,31,34,24,33);(7,10)(8,15)(9,20)(11,25)(12,27)(13,26)(14,28)(16,24)(17,32)(18,31)(19,33)(21,35)(22,37)(23,36)(29,38)(30,34)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6280104);