Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1902 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#10952 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2778 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)(9,14,20,29)(10,22)(11,18)(12,17)(13,16)(15,24)(19,33,35,36)(21,32,26,28)(23,30)(25,31)(27,37,38,34);(7,9,12,21,13,20,11,26)(8,14,18,28,16,29,17,32)(10,19,25,34,23,35,24,37)(15,27,30,33,31,38,22,36);(3,4,5,6)(7,10,11,24,13,23,12,25)(8,15,17,22,16,31,18,30)(9,19,26,37,20,35,21,34)(14,27,32,36,29,38,28,33)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261316);