Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#13285 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C23 | C2×#102 |
| prime radical | C24 | C2×SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
894 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(3,4)(5,9)(6,7)(8,10)(11,12,17,22,18,23,16,33)(13,24,28,40,29,41,27,36)(14,21,15,32,31,35,30,20)(19,34,37,42,38,25,39,26);(4,7)(8,9)(11,13,15,26,17,28,31,34,18,29,30,42,16,27,14,25)(12,19,35,41,33,39,32,40,23,38,21,24,22,37,20,36);(3,5,6,10)(4,8,7,9)(11,14,16,30,18,31,17,15)(12,20,22,21,23,32,33,35)(13,25,27,42,29,34,28,26)(19,36,37,24,38,40,39,41)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402575);