Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9091 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1869 |
| prime radical | C22×D8 | C8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
370 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 370 conjugacy classes of subgroups — too many to draw. Only its 100 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:
(1,2,7,10)(3,6,9,5)(4,11)(8,12)(13,15)(14,16)(17,18,21,24,22,25,29,31,20,19,23,26,28,27,30,32);(1,3,5,2,7,9,6,10)(4,11,13,8,14,16,12,15)(17,19,21,26,22,27,29,32,20,18,23,24,28,25,30,31);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,16)(10,15)(18,19)(24,26)(25,27)(31,32)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=g,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,c^b=ci,e^b=eg,d^c=dg,e^c=eg,f^c=fg92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 45037);