Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#11297 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | #385 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1622 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);(7,8,12,17,13,18,11,16)(9,19,22,14,23,29,21,30)(10,24,25,15,27,31,26,32)(20,34,35,38,36,28,37,33);(3,4)(5,6)(7,9,11,22,13,23,12,21)(8,14,16,29,18,30,17,19)(10,20,26,35,27,36,25,37)(15,28,24,33,32,34,31,38);(4,6)(7,10,11,25,13,27,12,26)(8,15,16,31,18,32,17,24)(9,20,21,35,23,36,22,37)(14,28,19,33,30,34,29,38)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265461);