Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C2×D8 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
652 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)(5,6,8,7)(13,14,16,20)(15,18,24,26)(17,25,23,21)(19,28,27,22);(2,4)(9,10,11,12)(14,18)(15,22)(17,23)(19,21)(20,26)(24,28)(25,27);(5,7,8,6)(14,18)(15,22)(17,23)(19,21)(20,26)(24,28)(25,27)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fi,d^2=hi,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,d^b=dh,e^b=eh,h^b=hi,d^c=dh,e^c=eh,h^c=hi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72322658, #72322806, #72322816, … |
| maximal normal subgroup of | #72322658, #72322806, #72322816, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54444);