Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8491 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#8501 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #379 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#121 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1992 conjugacy classes of subgroups.
| minimal degree | μ(G) = 66 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 66 points:
(1,2);(3,4)(5,13,16,9)(6,17,19,26)(7,12)(8,11)(10,18,24,20)(14,29)(15,28,33,23)(21,25)(22,30)(27,32)(31,34)(35,36,40,45,41,46,39,44)(37,47,51,42,50,57,49,56)(38,43,52,58,53,59,54,60)(48,62,63,66,64,61,65,55);(3,5)(4,9)(6,14)(7,15)(8,16)(10,22)(11,13)(12,23)(17,34)(18,30)(19,31)(20,32)(21,33)(24,27)(25,28)(26,29)(35,37)(36,42)(38,48)(39,51)(40,49)(41,50)(43,55)(44,57)(45,47)(46,56)(52,65)(53,64)(54,63)(58,61)(59,66)(60,62);(3,6,8,19)(4,10,11,24)(5,14,16,31)(7,20,21,18)(9,22,13,27)(12,26,25,17)(15,32,33,30)(23,29,28,34)(35,38)(36,43)(37,48)(39,54)(40,52)(41,53)(42,55)(44,60)(45,58)(46,59)(47,61)(49,65)(50,64)(51,63)(56,66)(57,62)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 94702);