Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8736 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#11171 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #381 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2482 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);(7,8,11,16,12,15,25,30)(9,17,19,31,20,13,23,27)(10,14,22,29,21,28,24,32)(18,33,35,38,34,26,36,37);(3,4)(5,6)(7,9,12,20)(8,13,15,17)(10,18,21,34)(11,23,25,19)(14,26,28,33)(16,31,30,27)(22,36,24,35)(29,38,32,37);(4,6)(7,10)(8,14,15,28)(9,18)(11,24)(12,21)(13,26,17,33)(16,32,30,29)(19,36)(20,34)(22,25)(23,35)(27,38,31,37)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265056);