Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13304 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×D8 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
842 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(9,10,11,12)(14,21)(15,26)(16,18)(17,28)(19,22)(23,24)(25,27);(5,6)(7,8)(13,14)(15,16)(17,27)(18,21)(19,26)(20,25)(22,24)(23,28);(2,4)(6,8)(13,15,16,14,22,21,17,26,20,23,28,25,19,27,18,24)Polycyclic presentation:
2^9|a^2=g,c^2=d,d^2=hi,e^2=h,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fhi,h^a=hi,c^b=cf,d^b=dh,e^b=ehi,f^b=fhi,h^b=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72322452, #72322603, #72322616, … |
| maximal normal subgroup of | #72322452, #72322603, #72322616, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55721);