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×#61794 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C2×#3547 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6126 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)(6,9)(7,10)(8,13)(12,17)(14,16)(15,18)(19,20,24,29)(21,25,30,38)(22,27,33,41)(23,26,35,28)(31,37,43,48)(32,39,44,36)(34,40,46,42)(45,49,50,47);(3,5)(4,8)(6,12)(7,13)(9,15)(10,11)(14,18)(16,17)(19,21)(20,25)(22,31)(23,32)(24,30)(26,39)(27,37)(28,36)(29,38)(33,43)(34,45)(35,44)(40,49)(41,48)(42,47)(46,50);(3,6,7,14)(4,9,10,16)(5,12,13,18)(8,15,11,17)(20,26)(21,30)(23,35)(25,36)(27,40)(28,29)(31,43)(34,46)(37,47)(38,39)(41,42)(48,49);(5,13)(8,11)(12,18)(15,17)(19,22)(20,27)(21,31)(23,34)(24,33)(25,37)(26,40)(28,42)(29,41)(30,43)(32,45)(35,46)(36,47)(38,48)(39,49)(44,50);(4,10)(8,11)(9,16)(15,17)(21,30)(22,33)(25,38)(27,41)(32,44)(34,46)(36,39)(40,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10485166);