Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1902 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C2×C4 | C22×C16 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1869 |
| prime radical | C22×D4 | C2×C8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
350 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 350 conjugacy classes of subgroups — too many to draw. Only its 112 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)(3,6)(4,12,15,10)(5,11)(7,9)(8,14,16,13)(17,18,20,24,21,25,28,31,22,19,23,26,29,27,30,32);(1,3,6,2,7,11,5,9)(4,10,13,8,15,12,14,16)(17,19,20,26,21,27,28,32,22,18,23,24,29,25,30,31);(1,4,7,15)(2,8,9,16)(3,10,11,12)(5,14,6,13)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=g,d^2=g,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^b=eg,d^c=dg,e^c=eg,f^c=fg116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 45046);