Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#24798 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2526 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 38 points:
(1,2);(3,4,5,6)(7,8,12,18)(9,14,22,30)(10,23,26,32)(11,17,24,16)(13,21,28,20)(15,31,33,25)(19,34,37,27)(29,36,38,35);(7,9,11,20,12,22,24,21)(8,13,16,14,18,28,17,30)(10,19,25,35,26,37,31,36)(15,27,23,29,33,34,32,38);(4,6)(7,9)(8,14)(10,19)(11,20)(12,22)(13,17)(15,29)(16,28)(18,30)(21,24)(23,34)(25,35)(26,37)(27,32)(31,36)(33,38);(7,10,24,31,12,26,11,25)(8,15,17,32,18,33,16,23)(9,19,21,36,22,37,20,35)(13,27,30,38,28,34,14,29)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350142);