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×#49101 |
| 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)(11,12,15,16)(13,17,21,19)(14,22,23,18)(20,25,26,24)(27,28)(29,35)(30,38)(31,32)(33,37)(34,39)(36,40)(41,42);(3,4,7,6)(5,8,9,10)(12,16)(14,23)(17,19)(20,26)(28,32)(33,35)(34,38)(40,41);(3,5,7,9)(4,8,6,10)(11,12)(13,19)(14,18)(15,16)(17,21)(20,25)(22,23)(24,26)(27,29)(28,33)(30,36)(31,37)(32,35)(34,40)(38,41)(39,42);(3,6,7,4)(5,10,9,8)(11,13)(12,17)(14,20)(15,21)(16,19)(18,24)(22,25)(23,26)(27,28)(29,33)(30,34)(31,32)(35,37)(36,40)(38,39)(41,42);(11,14,15,23)(12,18,16,22)(13,20,21,26)(17,24,19,25)(27,30,31,39)(28,34,32,38)(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, 7612110);