Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#11285 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #124 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | #394 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1694 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)(9,13,18,25)(10,14,21,28)(11,16)(12,15)(17,24)(19,26,32,37)(20,27)(22,29,34,35)(23,30)(31,36)(33,38);(7,9)(8,13,15,25)(10,20,21,33)(11,19)(12,18)(14,27)(16,26,30,37)(17,22,31,34)(23,32)(24,35)(28,38)(29,36);(7,10)(8,14,15,28)(9,17,18,31)(11,22)(12,21)(13,24)(16,29,30,35)(19,20,32,33)(23,34)(25,36)(26,27)(37,38)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265443);