Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1343 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #99 |
| commutator | C22×C4 | C4×C8 |
| Frattini, cosocle | C22×C4⋊C8 | C22 |
| socle | C23 | #323 |
| prime radical | C24 | C22⋊C8 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C4 ◁ C22×C4⋊C8 ◁ G | |
| LECS | G ▷ C22×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
334 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 334 conjugacy classes of subgroups — too many to draw. Only its 82 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)(13,14,16,22,18,23,24,27,19,21,25,28,17,26,15,20);(1,2,3,4)(6,9,11,10)(13,15,17,25,19,24,18,16)(14,20,23,22,21,27,26,28)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=hi,d^2=g,e^2=i,g^2=i,b^a=bc,c^a=cf,e^a=eh,f^a=fh,c^b=ci,d^b=dfhi,d^c=dh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 189);