Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61629 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#61930 |
| 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 | |
6118 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)(5,13)(6,10)(7,11)(8,12)(9,15)(14,18)(16,17)(19,20)(21,26)(22,33)(23,28)(24,37)(25,29)(27,34)(30,45)(31,40)(32,41)(35,43)(36,42)(38,44)(39,48)(46,50)(47,49);(3,5,8,15)(4,9,12,13)(6,14,7,16)(10,17,11,18)(19,21)(20,26)(22,30)(23,36)(24,32)(25,31)(27,39)(28,42)(29,40)(33,45)(34,48)(35,46)(37,41)(38,47)(43,50)(44,49);(3,6,8,7)(4,10,12,11)(5,14,15,16)(9,17,13,18)(19,22)(20,27)(21,30)(23,34)(24,38)(25,35)(26,39)(28,43)(29,44)(31,46)(32,47)(33,37)(36,48)(40,49)(41,45)(42,50);(3,7,8,6)(4,11,12,10)(5,14,15,16)(9,17,13,18)(21,31)(26,40)(30,46)(32,36)(39,49)(41,42)(45,50)(47,48);(4,12)(9,13)(10,11)(17,18)(19,23,25,24)(20,28,29,37)(21,32,31,36)(22,34,35,38)(26,41,40,42)(27,43,44,33)(30,47,46,48)(39,45,49,50)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10485803);