Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61643 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#61672 |
| 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 | |
7514 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)(5,9)(6,8)(7,10)(11,12,16,20)(13,17,22,32)(14,24,27,36)(15,28,30,19)(18,29,35,26)(21,37,25,42)(23,40,39,33)(31,41,34,38);(3,5)(4,7)(6,10)(8,9)(11,13)(12,17)(14,25)(15,23)(16,22)(18,34)(19,33)(20,32)(21,27)(24,42)(26,41)(28,40)(29,38)(30,39)(31,35)(36,37);(11,14,16,27)(12,18,20,35)(13,21,22,25)(15,29,30,26)(17,31,32,34)(19,36,28,24)(23,41,39,38)(33,42,40,37);(5,10)(7,9)(11,13)(12,17)(14,21)(15,23)(16,22)(18,31)(19,33)(20,32)(24,37)(25,27)(26,38)(28,40)(29,41)(30,39)(34,35)(36,42);(4,8)(7,9)(13,22)(17,32)(21,25)(23,39)(31,34)(33,40)(37,42)(38,41)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10484473);