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×#51313 |
| 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 | |
4530 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);(3,4,7,10)(5,8,6,9)(12,16)(17,20)(18,23)(24,26)(27,28,32,30)(29,33,37,35)(31,38,39,34)(36,41,42,40);(3,5,7,6)(4,8,10,9)(12,16)(13,19)(14,22)(24,26)(28,30)(33,35)(34,38)(40,41);(3,4)(5,8)(6,9)(7,10)(11,12)(13,20)(14,18)(15,16)(17,19)(21,26)(22,23)(24,25)(27,29,32,37)(28,33,30,35)(31,36,39,42)(34,40,38,41);(3,5,7,6)(4,8,10,9)(11,13)(12,17)(14,21)(15,19)(16,20)(18,24)(22,25)(23,26)(27,30,32,28)(29,35,37,33)(31,38,39,34)(36,41,42,40);(3,6,7,5)(4,9,10,8)(11,14,15,22)(12,18,16,23)(13,21,19,25)(17,24,20,26)(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, 7614015);