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×#56205 |
| 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 | |
4620 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)(12,16)(13,19)(18,23)(21,25)(27,28,32,35)(29,30,33,38)(31,34,40,42)(36,37,39,41);(4,5)(8,9)(14,22)(18,23)(21,25)(24,26)(27,29,32,33)(28,30,35,38)(31,39,40,36)(34,41,42,37);(3,4,7,5)(6,8,10,9)(11,12,15,16)(13,20,19,17)(14,18,22,23)(21,26,25,24)(27,29)(28,30)(31,36)(32,33)(34,37)(35,38)(39,40)(41,42);(3,5,7,4)(6,9,10,8)(11,13,15,19)(12,17,16,20)(14,21,22,25)(18,24,23,26)(27,30)(28,33)(29,35)(31,37)(32,38)(34,39)(36,42)(40,41);(3,6)(4,8)(5,9)(7,10)(11,14,15,22)(12,18,16,23)(13,21,19,25)(17,24,20,26)(27,31)(28,34)(29,36)(30,37)(32,40)(33,39)(35,42)(38,41)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8006189);