Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1655 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#8505 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | #379 |
| prime radical | C2×#3531 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2074 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 50 points:
(1,2);(4,8,10,13)(9,17,18,16)(19,20)(21,29,32,25)(22,33,34,44)(23,28)(24,27)(26,35,41,37)(30,46)(31,43,36,40)(38,42)(39,48)(45,49)(47,50);(3,4)(5,9)(6,13)(7,10)(8,15)(11,16)(12,18)(14,17)(19,21)(20,25)(22,30)(23,36)(24,32)(26,39)(27,29)(28,43)(31,38)(33,46)(34,47)(35,49)(37,48)(40,42)(41,45)(44,50);(3,5,7,12)(4,9,10,18)(6,14,15,11)(8,16,13,17)(19,22)(20,26)(21,30)(23,37)(24,34)(25,39)(27,41)(28,44)(29,45)(31,49)(32,47)(33,42)(35,38)(36,48)(40,46)(43,50)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 94773);