Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2230 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×#13187 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #476 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
702 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)(9,19,23,37)(10,25)(11,15)(12,30)(13,18)(14,21,32,20)(16,27)(17,26)(22,35,24,36)(28,33,38,31)(29,34);(7,9,12,22,13,23,26,24)(8,14,17,31,18,32,30,33)(10,20,11,28,27,21,29,38)(15,19,25,36,34,37,16,35);(3,4,5,6)(7,10,26,29,13,27,12,11)(8,15,30,16,18,34,17,25)(9,20,24,38,23,21,22,28)(14,19,33,35,32,37,31,36)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402272);