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×#54676 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4674 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)(5,9)(6,7)(8,10)(11,12,16,17)(13,18,21,24)(14,19,23,26)(15,22,25,20)(27,28,31,34)(29,35,37,32)(30,33,38,40)(36,41,42,39);(4,7)(8,9)(12,17)(13,21)(14,23)(15,25)(27,28,31,34)(29,32,37,35)(30,33,38,40)(36,39,42,41);(11,13)(12,18)(14,20)(15,19)(16,21)(17,24)(22,23)(25,26)(27,28)(29,32)(30,33)(31,34)(35,37)(36,39)(38,40)(41,42);(3,4)(5,8)(6,7)(9,10)(11,14)(12,19)(13,22)(15,24)(16,23)(17,26)(18,25)(20,21)(27,29,31,37)(28,32,34,35)(30,36,38,42)(33,39,40,41);(3,5,6,10)(4,8,7,9)(11,15,16,25)(12,20,17,22)(13,19,21,26)(14,24,23,18)(27,30,31,38)(28,33,34,40)(29,36,37,42)(32,39,35,41)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8010872);