Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13331 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#13446 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #459 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1042 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)(9,21)(10,28)(11,18)(12,17)(13,30)(14,33)(15,20)(16,23)(19,31)(22,24)(25,37)(26,29)(27,38)(32,36)(34,35);(7,9,15,27)(8,16,20,37)(10,29,17,35)(11,24,13,32)(12,23,28,25)(14,34,31,26)(18,21,30,38)(19,22,33,36);(4,6)(7,10,11,19,31,18,28,8,15,17,13,33,14,30,12,20)(9,22,23,29,34,37,32,38,27,36,25,35,26,16,24,21)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406377);