Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#11237 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #124 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | #400 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1778 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);(4,5)(7,8,12,16,13,17,11,27)(9,18,21,30,22,29,20,14)(10,23,25,32,26,38,24,15)(19,28,35,33,36,31,34,37);(3,4,6,5)(7,9,11,20,13,22,12,21)(8,14,16,18,17,30,27,29)(10,19,24,34,26,36,25,35)(15,31,23,37,32,28,38,33);(3,5,6,4)(7,10,11,24,13,26,12,25)(8,15,16,23,17,32,27,38)(9,19,20,34,22,36,21,35)(14,28,18,33,30,31,29,37)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265246);