Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2190 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#13026 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #447 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1350 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)(7,8,11,16,26,35,27,36,13,18,10,15,12,17,28,37)(9,19,21,31,29,34,24,14,23,30,20,38,22,33,25,32);(3,4,5,6)(7,9)(8,14)(10,24)(11,25)(12,29)(13,23)(15,33)(16,34)(17,38)(18,32)(19,36)(20,27)(21,28)(22,26)(30,37)(31,35);(7,9)(8,14)(10,25)(11,24)(12,29)(13,23)(15,34)(16,33)(17,38)(18,32)(19,37)(20,28)(21,27)(22,26)(30,36)(31,35)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401729);