Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #118454 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #388 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1924 |
| prime radical | C22×D4 | M16 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
223 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 223 conjugacy classes of subgroups — too many to draw. Only its 70 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:
(2,6)(3,10)(4,7)(8,15)(9,13)(12,16)(17,18,21,26,20,19,23,28,22,25,27,32,29,24,31,30);(1,2,7,13,5,9,4,6)(3,8,14,15,10,16,11,12)(17,19,27,28,20,25,31,32,22,24,21,30,29,18,23,26);(1,3,4,11,5,10,7,14)(2,8,6,12,9,16,13,15)(18,24,25,19)(21,27)(23,31)(26,28,32,30)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=d,d^2=gi,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,c^b=ch,d^b=di,f^b=fi,e^c=egi,f^c=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47360);