Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25140 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#25265 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2126 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2);(3,4)(5,8)(6,7)(9,10)(11,12,16,21,17,22,15,20)(13,18,25,34,26,35,24,33)(14,27,29,37,30,38,31,19)(23,39,40,36,28,42,41,32);(4,7)(8,10)(11,13,16,25,17,26,15,24)(12,18,21,34,22,35,20,33)(14,28,29,41,30,23,31,40)(19,36,27,42,37,32,38,39);(3,5,6,9)(4,8,7,10)(11,14)(12,19)(13,23)(15,29)(16,31)(17,30)(18,32)(20,27)(21,38)(22,37)(24,40)(25,41)(26,28)(33,39)(34,42)(35,36);(4,7)(8,10)(11,15,17,16)(12,20,22,21)(13,24,26,25)(14,29,30,31)(18,33,35,34)(19,27,37,38)(23,40,28,41)(32,39,36,42)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6348888);