Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C29 |
| direct product | C2×#10753 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | #385 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4034 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2);(3,4,5,6)(7,8,11,17,13,18,12,16)(9,19,23,28,22,29,21,14)(10,15,25,30,26,32,24,31)(20,34,35,27,36,33,37,38);(7,9,12,21,13,22,11,23)(8,14,16,29,18,28,17,19)(10,20,24,37,26,36,25,35)(15,27,31,34,32,38,30,33);(7,10)(8,15)(9,20)(11,25)(12,24)(13,26)(14,27)(16,31)(17,30)(18,32)(19,33)(21,37)(22,36)(23,35)(28,38)(29,34)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260641);