Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #466 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #9554 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ C4×M32 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
256 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 256 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,9,7,13,21,12,8,14,5,22,6,24,20,11)(3,15,16,10,18,25,31,26,19,28,23,27,17,32,30,29);(2,9,13,12,14,22,24,11)(10,25,26,28,27,32,29,15);(1,3,4,16,7,18,21,31,8,19,5,23,6,17,20,30)(2,10,11,15,24,29,22,32,14,27,12,28,13,26,9,25)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=d,d^2=gi,e^2=gi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72320529, #72320710, #72320764 |
| maximal normal subgroup of | #72320529, #72320710, #72320764 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47374);