Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#51306 |
| 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 | |
4632 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);(5,9)(8,10)(11,12,16,13)(14,17,22,19)(15,20,23,18)(21,25,26,24)(27,28)(29,35)(30,38)(31,32)(33,37)(34,39)(36,40)(41,42);(3,4,7,6)(5,8,9,10)(12,13)(15,23)(17,19)(21,26)(28,32)(33,35)(34,38)(40,41);(3,5)(4,8)(6,10)(7,9)(11,13,16,12)(14,17,22,19)(15,20,23,18)(21,24,26,25)(27,29,31,37)(28,33,32,35)(30,36,39,42)(34,40,38,41);(3,4,7,6)(5,8,9,10)(11,14)(12,17)(13,19)(15,21)(16,22)(18,24)(20,25)(23,26)(27,28)(29,33)(30,34)(31,32)(35,37)(36,40)(38,39)(41,42);(3,6,7,4)(5,10,9,8)(11,15,16,23)(12,18,13,20)(14,21,22,26)(17,24,19,25)(27,30)(28,34)(29,36)(31,39)(32,38)(33,40)(35,41)(37,42)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7614432);