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×#10752 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | #379 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4178 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);(7,8,12,16,13,17,11,24)(9,18,20,14,21,29,25,28)(10,15,22,31,23,32,26,30)(19,34,37,38,36,33,35,27);(3,4,5,6)(7,9)(8,14)(10,19)(11,25)(12,20)(13,21)(15,27)(16,29)(17,28)(18,24)(22,37)(23,36)(26,35)(30,33)(31,34)(32,38);(7,10)(8,15)(9,19)(11,26)(12,22)(13,23)(14,27)(16,31)(17,32)(18,33)(20,37)(21,36)(24,30)(25,35)(28,38)(29,34)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260639);