Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61049 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#61412 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6850 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);(4,8)(9,10)(11,12)(13,22)(14,26)(15,20)(16,21)(17,30)(18,24)(19,27)(23,38)(25,34)(28,35)(29,33)(31,36)(32,40)(37,41)(39,42);(3,4,7,8)(5,9,6,10)(11,13,16,24)(12,18,21,22)(14,23,28,39)(15,29,17,25)(19,32,36,37)(20,34,30,33)(26,42,35,38)(27,41,31,40);(3,5,7,6)(4,9,8,10)(11,14,16,28)(12,19,21,36)(13,23,24,39)(15,27,17,31)(18,32,22,37)(20,35,30,26)(25,40,29,41)(33,42,34,38);(3,6,7,5)(4,9,8,10)(13,24)(18,22)(23,39)(25,29)(32,37)(33,34)(38,42)(40,41);(11,15,16,17)(12,20,21,30)(13,25,24,29)(14,27,28,31)(18,33,22,34)(19,35,36,26)(23,40,39,41)(32,42,37,38)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10483895);