Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60771 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212×C4 |
| direct product | C2×#60828 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
7518 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 42 points:
(1,2);(3,4)(5,9)(6,8)(7,10)(11,12)(13,21)(14,25)(15,17)(16,29)(18,24)(19,26)(20,28)(22,36)(23,34)(27,33)(30,31)(32,42)(35,37)(38,39)(40,41);(3,5,6,10)(4,7,8,9)(12,17)(18,31)(19,25)(20,29)(21,34)(32,39)(33,40)(35,36);(5,10)(7,9)(11,13,15,23)(12,18,17,31)(14,22,26,37)(16,30,28,24)(19,32,25,39)(20,34,29,21)(27,42,41,38)(33,36,40,35);(4,8)(7,9)(11,14)(12,19)(13,22)(15,26)(16,27)(17,25)(18,32)(20,33)(21,35)(23,37)(24,38)(28,41)(29,40)(30,42)(31,39)(34,36);(14,26)(19,25)(22,37)(27,41)(32,39)(33,40)(35,36)(38,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482731);