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×#61627 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
7622 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)(13,18)(14,26)(15,20)(16,29)(17,21)(19,27)(22,37)(23,32)(24,33)(25,34)(28,35)(30,36)(31,38)(39,42)(40,41);(3,5)(4,7)(6,10)(8,9)(11,13)(12,18)(14,22)(15,23)(16,24)(17,25)(19,31)(20,32)(21,34)(26,37)(27,38)(28,40)(29,33)(30,39)(35,41)(36,42);(11,14,17,28)(12,19,21,36)(13,22,25,40)(15,27,16,30)(18,31,34,42)(20,35,29,26)(23,38,24,39)(32,41,33,37);(5,10)(7,9)(11,15,17,16)(12,20,21,29)(13,23,25,24)(14,27,28,30)(18,32,34,33)(19,35,36,26)(22,38,40,39)(31,41,42,37);(4,8)(7,9)(11,15,17,16)(12,20,21,29)(13,24,25,23)(14,27,28,30)(18,33,34,32)(19,35,36,26)(22,39,40,38)(31,37,42,41)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10484423);