Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2986 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(6,8)(9,10,11,12)(13,14,19,22)(15,23,25,27)(16,20,26,24)(17,21,18,28);(2,4)(13,15,18,24,19,25,17,20)(14,16,21,23,22,26,28,27);(5,6,7,8)(10,12)(13,16)(14,20)(15,21)(17,27)(18,23)(19,26)(22,24)(25,28);(13,17,19,18)(14,21,22,28)(15,20,25,24)(16,23,26,27)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=fgi,d^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,d^c=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72315669, #72315694, #72315742, … |
| maximal normal subgroup of | #72315669, #72315694, #72315742, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261275);