Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30490 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#30494 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#454 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3138 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)(7,8,13,14,12,18,11,17)(9,19,27,30,22,15,20,28)(10,23,26,16,25,32,24,31)(21,35,34,33,37,38,36,29);(8,14)(9,20)(10,24)(11,13)(16,31)(17,18)(22,27)(25,26)(28,30)(29,38)(33,35)(34,36);(4,6)(7,9,12,22)(8,15,18,19)(10,21,25,37)(11,20,13,27)(14,28,17,30)(16,29,31,33)(23,38,32,35)(24,36,26,34);(4,6)(7,10)(8,16)(9,21)(11,26)(12,25)(13,24)(14,23)(15,29)(17,32)(18,31)(19,33)(20,34)(22,37)(27,36)(28,38)(30,35)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408101);