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×#57828 |
| 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)(5,7)(6,9)(8,10)(11,12,16,14)(13,17,19,21)(15,22,23,18)(20,26,25,24)(27,28)(29,31)(30,37)(32,34)(33,35)(36,39)(38,41)(40,42);(4,5)(6,10)(12,14)(13,19)(15,23)(24,26)(27,29,33,30)(28,31,35,37)(32,38,40,36)(34,41,42,39);(4,5)(8,9)(11,13)(12,17)(14,21)(15,20)(16,19)(18,24)(22,26)(23,25)(27,30,33,29)(28,31,35,37)(32,38,40,36)(34,39,42,41);(3,5,7,4)(6,9,10,8)(11,14,16,12)(13,17,19,21)(15,22,23,18)(20,24,25,26)(27,31)(28,30)(29,35)(32,39)(33,37)(34,38)(36,42)(40,41);(3,6)(4,8)(5,9)(7,10)(11,15,16,23)(12,18,14,22)(13,20,19,25)(17,24,21,26)(27,32,33,40)(28,34,35,42)(29,36,30,38)(31,39,37,41)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8008708);