Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #134691 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
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 | #1835 |
| 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 | |
184 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)(19,31)(20,38,41,69)(21,43)(22,27)(23,46)(24,30)(25,34)(26,40,52,72)(28,32)(29,44)(33,61,64,47)(35,56)(36,48)(37,58)(39,70,60,53)(42,73,71,51)(45,54)(49,67)(50,59)(55,63,76,80)(57,66)(62,68,78,77)(65,74,79,75);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19,32,59,44,66,22,35,24,37,21,34,23,36,45,67)(18,25,27,31,46,56,28,48,30,50,54,58,29,49,43,57)(20,33,60,78,71,79,40,63,41,64,39,62,42,65,72,80)(26,47,51,55,70,75,38,68,52,61,73,76,53,74,69,77);(3,9)(6,12)(10,15)(13,16)(17,20,21,39,44,71,45,72,24,41,32,60,23,42,22,40)(18,26,27,51,46,70,28,38,30,52,54,73,29,53,43,69)(19,33,34,62,66,79,67,80,37,64,59,78,36,65,35,63)(25,47,31,55,56,75,48,68,50,61,58,76,49,74,57,77)Polycyclic presentation:
2^9|a^2=f,b^2=eg,c^2=d,d^2=gi,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,c^b=ch,f^c=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42875);