Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33607 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#33703 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1638 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2);(3,4)(5,8)(6,7)(9,10)(11,12,18,23)(13,24,26,25)(14,30,20,16)(15,31,33,21)(17,34,32,22)(19,35,36,27)(28,38,41,39)(29,37,42,40);(4,7)(8,10)(11,13,16,28,17,29,15,27,18,26,30,41,32,42,33,35)(12,19,21,37,22,38,20,24,23,36,31,40,34,39,14,25);(3,5,6,9)(4,8,7,10)(11,14,15,22,32,31,16,12,18,20,33,34,17,21,30,23)(13,25,27,38,42,40,28,19,26,24,35,39,29,37,41,36);(13,26)(19,36)(24,25)(27,35)(28,41)(29,42)(37,40)(38,39)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530362);