Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9974 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #392 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | #1444 |
| prime radical | C24 | C4×Q8 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
460 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 460 conjugacy classes of subgroups — too many to draw. Only its 142 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,7)(3,10)(4,5)(8,15)(9,14)(12,13)(17,18,22,26,23,27,21,29)(19,28,31,32,25,20,30,24);(1,2,4,7,6,9,5,14)(3,11,12,15,13,16,10,8)(17,19,21,30,23,25,22,31)(18,24,26,28,27,32,29,20);(1,3,5,10,6,13,4,12)(2,8,7,11,9,15,14,16)(17,20,21,32,23,28,22,24)(18,25,26,30,27,19,29,31)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=e,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,c^b=cf,e^b=eh,f^b=fhi,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 28569);