Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#57452 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2828 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4810 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);(11,12,15,18)(13,19,20,16)(14,17,22,24)(21,25,26,23)(27,28,31,32)(29,33,35,39)(30,34,37,40)(36,41,38,42);(3,4,7,9)(5,6,10,8)(13,20)(16,19)(21,26)(23,25)(28,32)(29,35)(30,37)(41,42);(3,5,7,10)(4,8,9,6)(11,13)(12,16)(14,21)(15,20)(17,23)(18,19)(22,26)(24,25)(27,29)(28,33)(30,38)(31,35)(32,39)(34,42)(36,37)(40,41);(3,5)(4,6)(7,10)(8,9)(11,12)(13,16)(14,17)(15,18)(19,20)(21,23)(22,24)(25,26)(27,30)(28,34)(29,36)(31,37)(32,40)(33,41)(35,38)(39,42);(3,6)(4,5)(7,8)(9,10)(11,14,15,22)(12,17,18,24)(13,21,20,26)(16,23,19,25)(27,28)(29,33)(30,34)(31,32)(35,39)(36,41)(37,40)(38,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8008220);