Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2048 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#11188 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #394 |
| 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 | |
2466 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)(7,8,12,17,13,16,11,28)(9,18,21,34,22,14,20,30)(10,23,26,32,27,33,25,15)(19,31,24,29,37,38,36,35);(7,9,13,22)(8,14,16,18)(10,24,27,36)(11,21,12,20)(15,31,32,38)(17,34,28,30)(19,25,37,26)(23,35,33,29);(4,6)(7,10)(8,15,16,32)(9,19)(11,25)(12,26)(13,27)(14,29,18,35)(17,23,28,33)(20,36)(21,24)(22,37)(30,38,34,31)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265140);