Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60756 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#60761 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
8040 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)(11,12,17,21)(13,18,24,33)(14,26,28,34)(15,20,16,29)(19,30,35,27)(22,36,38,41)(23,32,25,39)(31,40,42,37);(3,5)(4,7)(6,10)(8,9)(11,13)(12,18)(14,22)(15,23)(16,25)(17,24)(19,31)(20,32)(21,33)(26,36)(27,37)(28,38)(29,39)(30,40)(34,41)(35,42);(5,10)(7,9)(11,14)(12,19)(13,22)(15,27)(16,30)(17,28)(18,31)(20,34)(21,35)(23,37)(24,38)(25,40)(26,29)(32,41)(33,42)(36,39);(4,8)(7,9)(11,15,17,16)(12,20,21,29)(13,23,24,25)(14,27,28,30)(18,32,33,39)(19,34,35,26)(22,37,38,40)(31,41,42,36);(13,24)(18,33)(22,38)(23,25)(31,42)(32,39)(36,41)(37,40)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482622);