Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24398 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#24675 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2238 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)(5,9)(6,7)(8,10)(11,12,16,21,15,20,17,22)(13,23,26,38,25,18,32,34)(14,27,30,36,29,37,31,19)(24,40,28,33,41,35,42,39);(4,7)(8,9)(11,13)(12,18)(14,28)(15,25)(16,26)(17,32)(19,35)(20,23)(21,34)(22,38)(24,31)(27,39)(29,42)(30,41)(33,37)(36,40);(11,14,15,29)(12,19,20,36)(13,24,25,41)(16,31,17,30)(18,33,23,39)(21,37,22,27)(26,42,32,28)(34,40,38,35);(3,5,6,10)(4,8,7,9)(12,20)(18,23)(19,36)(21,22)(27,37)(33,39)(34,38)(35,40)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6356199);