Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C27 |
| direct product | C2×#12980 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C2×D4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C23 | C8×D4 |
| prime radical | C22×#135 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ C23×C16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1308 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);(4,6)(7,8,12,18,13,19,32,23,14,20,33,37,11,17,22,35)(9,21,16,36,24,10,26,28,25,27,30,38,29,15,34,31);(3,4,5,6)(7,9,22,26,11,29,33,16,14,25,32,34,13,24,12,30)(8,15,35,36,17,27,37,31,20,10,23,38,19,21,18,28);(7,10)(8,16)(9,23)(11,21)(12,31)(13,27)(14,15)(17,34)(18,29)(19,26)(20,30)(22,38)(24,37)(25,35)(28,33)(32,36)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401607);