Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #134691 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #427 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1846 |
| prime radical | C23 | #427 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #376 ◁ #1726 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
204 conjugacy classes of subgroups.
| minimal degree | μ(G) = 80 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 80 points:
(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18)(19,30,34,56)(20,36,40,68)(21,42)(22,28)(23,46)(24,29)(25,32,49,63)(26,39,52,41)(27,44)(31,57)(33,55,35,65)(37,69)(38,71,58,51)(43,53)(45,74)(47,76)(48,62,75,64)(50,78)(54,61)(59,66)(60,79)(67,72,80,73)(70,77);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19,21,32,23,35,44,64,24,34,45,63,22,33,43,62)(18,25,27,30,28,48,42,55,29,49,53,56,46,75,74,65)(20,37,38,70,41,60,73,76,40,31,58,59,39,61,72,78)(26,50,51,79,36,66,67,69,52,47,71,54,68,77,80,57);(3,9)(6,12)(10,15)(13,16)(17,20)(18,26,29,52)(19,31)(21,38)(22,39)(23,41)(24,40)(25,47,49,50)(27,51,53,71)(28,36,46,68)(30,54,56,79)(32,59)(33,60)(34,37)(35,61)(42,67,74,80)(43,72)(44,73)(45,58)(48,77,75,66)(55,57,65,69)(62,76)(63,70)(64,78)Polycyclic presentation:
2^9|a^2=f,b^2=d,c^2=eg,d^2=g,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=chi,f^b=fi,f^c=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42950);