Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13304 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C2×D8 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
616 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(6,7)(13,14,17,22,15,23,24,28,16,21,25,27,18,19,26,20);(5,6,8,7)(9,10)(11,12)(13,15,16,18)(14,19,21,23)(17,24,25,26)(20,27,28,22);(2,4)(5,7,8,6)(10,12)(14,20)(15,18)(17,26)(19,22)(21,28)(23,27)(24,25)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=di,e^2=hi,g^2=hi,h^2=i,b^a=bd,c^a=ce,c^b=cf,e^c=eh,g^c=gh,h^c=hi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56031);