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×#51305 |
| 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 | |
4600 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)(13,16)(14,17)(15,20)(18,23)(19,22)(21,25)(24,26)(27,28,32,30)(29,33,37,35)(31,38,39,34)(36,41,42,40);(3,4,7,6)(5,8,9,10)(12,13)(15,23)(17,19)(21,26)(28,30)(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)(27,29,32,37)(28,33,30,35)(31,36,39,42)(34,40,38,41);(3,4,7,6)(5,8,9,10)(11,14,16,22)(12,17,13,19)(15,21,23,26)(18,24,20,25)(27,30,32,28)(29,35,37,33)(31,38,39,34)(36,41,42,40);(3,6,7,4)(5,10,9,8)(11,15)(12,18)(13,20)(14,21)(16,23)(17,24)(19,25)(22,26)(27,31,32,39)(28,34,30,38)(29,36,37,42)(33,40,35,41)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7614176);