Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135024 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1802 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C22×D8 | Q8 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
306 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 306 conjugacy classes of subgroups — too many to draw. Only its 90 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:
(1,2,5,8,6,9,4,14)(3,10,13,15,11,16,12,7)(17,18,21,25,19,22,28,26)(20,29,31,24,27,23,30,32);(1,3,4,12,6,11,5,13)(2,7,8,10,9,15,14,16)(18,22)(23,29)(24,32)(25,26);(3,11)(7,15)(10,16)(12,13)(18,23)(20,30)(21,28)(22,29)(24,25)(26,32)(27,31)Polycyclic presentation:
2^9|a^2=f,b^2=di,d^2=g,e^2=h,f^2=gi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=cg,e^c=ehi,f^c=fi,h^c=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72319563, #72319582, #72319670, … |
| maximal normal subgroup of | #72319563, #72319582, #72319670, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42264);