Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #140220 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1926 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #3169 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
252 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 252 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:
(1,2,5,9,6,10,4,7)(3,11,14,15,13,8,12,16)(17,18,21,25,22,27,19,23)(20,29,31,26,30,24,28,32);(2,7,10,9)(8,15,11,16)(18,23,27,25)(20,30)(24,32,29,26)(28,31);(1,3,4,12,6,13,5,14)(2,8,9,16,10,11,7,15)(18,24)(19,21)(20,28)(23,32)(25,26)(27,29)(30,31)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=dh,d^2=g,e^2=h,f^2=g,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,e^b=ei,d^c=di,e^c=egh,f^c=fi,h^c=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72285410, #72285417, #72285422, … |
| maximal normal subgroup of | #72285410, #72285417, #72285422, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47505);