Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9091 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#11059 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2626 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)(7,8,12,18,13,16,11,17)(9,14,20,30,21,29,27,34)(10,22,25,32,26,15,24,33)(19,35,23,38,36,28,37,31);(3,4,5,6)(7,9)(8,14)(10,23)(11,27)(12,20)(13,21)(15,31)(16,29)(17,34)(18,30)(19,24)(22,38)(25,36)(26,37)(28,32)(33,35);(7,10,13,26)(8,15,16,22)(9,19,21,36)(11,25,12,24)(14,28,29,35)(17,33,18,32)(20,37,27,23)(30,38,34,31)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264902);