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×#51774 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4746 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);(3,4,7,5)(6,9,10,8)(11,12)(13,19)(14,22)(15,16)(17,21)(18,23)(20,24)(25,26)(27,28,32,33)(29,35,36,30)(31,39,40,34)(37,38,42,41);(3,5,7,4)(6,9,10,8)(12,16)(17,19)(18,22)(24,25)(28,33)(29,36)(31,40)(38,41);(3,6)(4,8)(5,9)(7,10)(11,13,15,21)(12,17,16,19)(14,20,23,26)(18,24,22,25)(27,29,32,36)(28,30,33,35)(31,37,40,42)(34,38,39,41);(11,12)(13,17)(14,18)(15,16)(19,21)(20,24)(22,23)(25,26)(27,30)(28,29)(31,38)(32,35)(33,36)(34,37)(39,42)(40,41);(11,14)(12,18)(13,20)(15,23)(16,22)(17,24)(19,25)(21,26)(27,31,32,40)(28,34,33,39)(29,37,36,42)(30,38,35,41)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7614968);