Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C26 |
| direct product | C2×#11470 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C23 | C2×C22⋊C8 |
| prime radical | C2×#525 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
1072 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(3,4)(5,9,10,7)(6,8)(11,12,16,22,17,23,30,37,18,24,14,20,15,21,31,38)(13,25,28,41,27,36,42,34,29,39,26,40,32,19,33,35);(3,5)(4,7)(6,10)(8,9)(11,13)(12,19)(14,28)(15,32)(16,26)(17,27)(18,29)(20,35)(21,39)(22,34)(23,25)(24,36)(30,33)(31,42)(37,40)(38,41);(11,14,17,31,18,16,15,30)(12,20,23,38,24,22,21,37)(13,26,27,33,29,28,32,42)(19,34,25,40,36,35,39,41)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387278);