Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1913 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#10940 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2794 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)(9,13,18,25)(10,14)(11,22)(12,15)(16,23)(17,29,30,24)(19,33,32,26)(20,28)(21,35)(27,34)(31,36,38,37);(7,9,11,19,12,18,23,32)(8,13,16,26,15,25,22,33)(10,17,21,31,20,30,34,38)(14,24,27,36,28,29,35,37);(3,4,5,6)(7,10,12,20)(8,14,15,28)(9,17,18,30)(11,21,23,34)(13,24,25,29)(16,27,22,35)(19,31,32,38)(26,36,33,37)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261300);