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 | |
604 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)(9,10,11,12)(13,14)(15,19)(16,18)(17,25)(20,24)(21,23)(22,26)(27,28);(5,6,8,7)(13,14,15,19)(16,18,22,26)(17,25,23,21)(20,28,27,24);(2,4)(5,7,8,6)(14,18)(16,24)(17,23)(19,26)(20,21)(22,28)(25,27)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=di,e^2=hi,f^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fh,h^a=hi,c^b=cf,e^b=eh,f^b=fh,h^b=hi,e^c=eh,f^c=fh,h^c=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72322452, #72322595, #72322609, … |
| maximal normal subgroup of | #72322452, #72322595, #72322609, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55946);