Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60457 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#60717 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6798 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 42 points:
(1,2);(3,4)(5,8)(6,7)(9,10)(11,12)(13,23)(14,26)(15,19)(16,21)(17,22)(18,32)(20,30)(24,38)(25,35)(27,40)(28,36)(29,41)(31,34)(33,42)(37,39);(4,7)(8,10)(11,13,17,15)(12,19,22,23)(14,24,29,27)(16,31,18,25)(20,33,36,37)(21,35,32,34)(26,40,41,38)(28,42,30,39);(11,14,18,30,17,29,16,28)(12,20,21,26,22,36,32,41)(13,24,25,39,15,27,31,42)(19,33,35,40,23,37,34,38);(3,5,6,9)(4,8,7,10)(11,15,17,13)(12,19,22,23)(14,27,29,24)(16,25,18,31)(20,33,36,37)(21,35,32,34)(26,40,41,38)(28,39,30,42);(11,16,17,18)(12,21,22,32)(13,25,15,31)(14,28,29,30)(19,34,23,35)(20,26,36,41)(24,39,27,42)(33,38,37,40)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482537);