Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61609 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#61801 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C2×#3551 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6180 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)(28,36)(30,38)(31,39)(32,37)(33,42)(34,43)(35,41)(40,46)(44,48)(45,49)(47,50);(3,5)(4,8)(6,12)(7,13)(9,15)(10,11)(14,18)(16,17)(19,21)(20,25)(22,33)(23,32)(24,31)(26,37)(27,42)(28,40)(29,39)(30,34)(35,47)(36,46)(38,43)(41,50)(44,49)(45,48);(3,6,7,14)(4,9,10,16)(5,12,13,18)(8,15,11,17)(20,26)(23,36)(25,37)(27,41)(28,29)(32,46)(35,48)(38,49)(39,40)(42,50)(43,44)(45,47);(5,13)(8,11)(12,18)(15,17)(19,22)(20,27)(21,30)(23,35)(24,34)(25,38)(26,41)(28,44)(29,43)(31,33)(32,45)(36,48)(37,49)(39,42)(40,50)(46,47);(4,10)(8,11)(9,16)(15,17)(21,31)(22,34)(25,39)(27,43)(32,46)(35,48)(37,40)(41,44)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10485174);