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×#60879 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6066 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 50 points:
(1,2);(3,4)(5,11)(6,9)(7,10)(8,13)(12,17)(14,16)(15,18)(19,20)(21,25)(22,27)(23,26)(24,29)(28,35)(30,37)(31,36)(32,39)(33,41)(34,40)(38,45)(42,46)(43,48)(44,47)(49,50);(3,5,7,13)(4,8,10,11)(6,14,15,12)(9,17,18,16)(19,21,24,32)(20,25,29,39)(22,30,33,43)(23,31,35,45)(26,36,28,38)(27,37,41,48)(34,44,46,50)(40,47,42,49);(3,6,7,15)(4,9,10,18)(5,12,13,14)(8,16,11,17)(20,26)(23,35)(25,36)(27,40)(28,29)(31,45)(34,46)(37,47)(38,39)(41,42)(44,50)(48,49);(4,10)(8,11)(9,18)(16,17)(19,22)(20,27)(21,30)(23,34)(24,33)(25,37)(26,40)(28,42)(29,41)(31,44)(32,43)(35,46)(36,47)(38,49)(39,48)(45,50);(22,33)(27,41)(30,43)(34,46)(37,48)(40,42)(44,50)(47,49)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482918);