Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8510 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #379 |
| commutator | C42 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1779 |
| prime radical | #3531 | C4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
519 conjugacy classes of subgroups.
| minimal degree | μ(G) = 80 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 80 points:
(2,6)(4,11)(7,15)(8,12)(9,13)(14,16)(17,18,22,29,24,30,46,59,23,27,45,53,21,28,42,56)(19,31,34,57,33,54,66,78,35,25,43,49,40,48,70,75)(20,36,39,68,38,52,69,58,37,55,44,71,41,26,50,51)(32,61,64,79,65,60,77,76,62,73,67,80,63,47,72,74);(1,2)(3,7)(4,12)(5,8)(6,11)(9,14)(10,16)(13,15)(17,19)(18,25)(20,32)(21,40)(22,43)(23,35)(24,33)(26,47)(27,31)(28,54)(29,57)(30,48)(34,45)(36,61)(37,62)(38,65)(39,67)(41,63)(42,66)(44,64)(46,70)(49,53)(50,77)(51,76)(52,60)(55,73)(56,75)(58,74)(59,78)(68,80)(69,72)(71,79);(1,3,5,10)(2,7,8,16)(4,13,11,9)(6,14,12,15)(17,20,23,37)(18,26,27,52)(19,32,35,62)(21,41,24,38)(22,44,45,39)(25,47,31,60)(28,55,30,36)(29,58,53,51)(33,65,40,63)(34,67,43,64)(42,69,46,50)(48,61,54,73)(49,76,57,74)(56,68,59,71)(66,72,70,77)(75,80,78,79)Polycyclic presentation:
2^9|a^2=f,c^2=g,d^2=gi,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dgi,e^b=eg,f^b=fi,d^c=dg,e^c=ei,f^c=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36874);