Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1408 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #353 |
| commutator | C42 | C4×C8 |
| Frattini, cosocle | C2×#384 | C22 |
| prime radical | C24 | #99 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C23×C4 ◁ C2×#384 ◁ G | |
| LECS | G ▷ C2×#384 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
366 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 366 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 128 |
A minimal faithful permutation representation — 2 generators on 28 points:
(5,6,7,9,8,10,11,12)(13,14)(15,22,23,18)(16,20,25,19)(17,21)(24,26,28,27);(1,2,3,4)(13,15,16,23,17,24,25,28)(14,18,19,22,21,27,20,26)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=hi,d^2=g,e^2=i,f^2=i,b^a=bc,c^a=cf,e^a=ehi,f^a=fh,h^a=hi,d^b=dfh,d^c=dhi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1297);