Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #124 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C23 | #397 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1474 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,4)(6,7)(9,10)(11,21)(12,26,30,35)(13,19)(14,18)(15,20)(16,23)(17,28,36,29)(22,31,38,40)(24,33)(25,32)(27,34,37,39);(5,6)(7,8)(9,11,14,24,15,25,13,23)(10,16,19,32,20,33,18,21)(12,27,28,22,30,37,29,38)(17,34,35,40,36,39,26,31);(2,4)(5,7,8,6)(9,12,13,28,15,30,14,29)(10,17,18,35,20,36,19,26)(11,22,23,37,25,38,24,27)(16,31,21,34,33,40,32,39)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=d,d^2=i,e^2=i,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,h^a=hi,c^b=cf,f^c=fi,g^c=gi,h^c=hi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30985);