Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#53153 |
| 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 | |
4614 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);(4,7)(5,9)(11,12)(13,16)(14,17)(15,20)(18,23)(19,21)(22,25)(24,26)(31,39)(33,40)(36,42)(38,41);(4,7)(8,10)(12,13)(14,21)(18,20)(22,26)(27,28,32,34)(29,35,37,30)(31,33,39,40)(36,41,42,38);(3,4,6,7)(5,8,9,10)(11,13,16,12)(14,17,21,19)(15,20,23,18)(22,24,26,25)(27,29,32,37)(28,30,34,35)(31,36,39,42)(33,38,40,41);(3,5)(4,8)(6,9)(7,10)(11,14,16,21)(12,17,13,19)(15,22,23,26)(18,24,20,25)(27,30)(28,29)(31,38)(32,35)(33,36)(34,37)(39,41)(40,42);(11,15,16,23)(12,18,13,20)(14,22,21,26)(17,24,19,25)(27,31,32,39)(28,33,34,40)(29,36,37,42)(30,38,35,41)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8000227);