Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3288 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
| direct product | C2×#22465 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#578 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4530 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)(7,8,12,18,14,10,13,17)(9,19,23,20,24,31,22,15)(11,16,28,33,29,25,27,34)(21,35,26,36,38,32,37,30);(7,9)(8,15)(10,20)(11,26)(12,23)(13,22)(14,24)(16,32)(17,31)(18,19)(21,27)(25,35)(28,38)(29,37)(30,33)(34,36);(4,6)(7,10,14,8)(9,20,24,15)(11,25,29,16)(12,18,13,17)(19,22,31,23)(21,36,38,30)(26,35,37,32)(27,34,28,33);(7,11)(8,16)(9,21)(10,25)(12,28)(13,27)(14,29)(15,30)(17,34)(18,33)(19,35)(20,36)(22,37)(23,26)(24,38)(31,32)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6286396);