Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9109 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
| direct product | C2×#10974 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4342 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)(7,8,11,16,12,15,24,30)(9,17,19,31,20,13,23,26)(10,14,22,29,21,28,34,38)(18,27,33,37,32,25,35,36);(7,9,12,20)(8,13,15,17)(10,18,21,32)(11,23,24,19)(14,27,28,25)(16,31,30,26)(22,35,34,33)(29,36,38,37);(4,6)(7,10)(8,14)(9,18)(11,22)(12,21)(13,25)(15,28)(16,29)(17,27)(19,33)(20,32)(23,35)(24,34)(26,36)(30,38)(31,37)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264641);