Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C16 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #99 |
| commutator | C2×C4 | C4×C16 |
| Frattini, cosocle | C22×C4⋊C8 | C22 |
| socle | C23 | #323 |
| prime radical | C24 | C22⋊C8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C4 ◁ C22×C4⋊C8 ◁ G | |
| LECS | G ▷ C22×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
348 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 348 conjugacy classes of subgroups — too many to draw. Only its 94 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 28 points:
(5,6)(7,10,12,9)(8,11)(13,14,16,21,18,23,26,28,19,24,25,27,17,22,15,20);(1,2,3,4)(6,9,11,10)(13,15,17,25,19,26,18,16)(14,20,22,27,24,28,23,21)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=h,d^2=g,e^2=i,g^2=i,b^a=bc,c^a=cf,e^a=eh,f^a=fh,d^b=dfh,d^c=dh104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 165);