Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61609 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#61681 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
7568 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)(13,19)(14,25)(15,20)(16,29)(17,23)(18,27)(21,28)(22,37)(24,32)(26,36)(30,34)(31,41)(33,39)(35,40)(38,42);(3,5)(4,7)(6,10)(8,9)(11,13)(12,17)(14,22)(15,23)(16,24)(18,31)(19,20)(21,32)(25,42)(26,38)(27,39)(28,40)(29,35)(30,41)(33,34)(36,37);(11,14,15,26)(12,18,20,34)(13,22,23,38)(16,30,28,27)(17,31,19,33)(21,36,29,25)(24,41,40,39)(32,37,35,42);(5,10)(7,9)(11,13)(12,19)(14,22)(15,23)(16,24)(17,20)(18,33)(21,35)(25,37)(26,38)(27,39)(28,40)(29,32)(30,41)(31,34)(36,42);(4,8)(7,9)(12,20)(13,23)(18,34)(21,29)(22,38)(24,40)(25,36)(39,41)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10484481);