Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60768 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#61009 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6184 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 50 points:
(1,2);(3,4)(5,11)(6,10)(7,9)(8,12)(13,17)(14,16)(15,18)(19,20)(21,25)(22,27)(23,26)(24,29)(28,36)(30,38)(31,37)(32,40)(33,42)(34,43)(35,41)(39,46)(44,48)(45,49)(47,50);(3,5)(4,8)(6,13)(7,12)(9,11)(10,15)(14,18)(16,17)(19,21,24,32)(20,25,29,40)(22,33,34,30)(23,31,36,46)(26,37,28,39)(27,42,43,38)(35,47,48,45)(41,50,44,49);(5,12)(8,11)(13,18)(15,17)(20,26)(23,36)(25,37)(27,41)(28,29)(31,46)(35,48)(38,49)(39,40)(42,50)(43,44)(45,47);(4,9)(8,11)(10,16)(15,17)(19,22,24,34)(20,27,29,43)(21,30,32,33)(23,35,36,48)(25,38,40,42)(26,41,28,44)(31,45,46,47)(37,49,39,50);(3,6,7,14)(4,10,9,16)(5,13,12,18)(8,15,11,17)(22,34)(27,43)(30,33)(35,48)(38,42)(41,44)(45,47)(49,50)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10483053);