Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #93047 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1294 |
| commutator | C22×C8 | C42 |
| Frattini, cosocle | C22×C4⋊C8 | C22 |
| prime radical | C24 | #102 |
| LCS | G ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×C4⋊C8 ◁ G | |
| LECS | G ▷ C22×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
325 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 325 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 24 points:
(2,5)(10,15,16,23)(11,18)(12,13)(17,24,22,19)(20,21);(1,2,3,6,4,5,7,8)(9,10)(11,19,20,24)(12,22)(13,17)(14,16)(15,21,23,18)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=f,e^2=i,f^2=h,g^2=h,b^a=bc,c^a=cf,e^a=efg,f^a=fh,g^a=gh,c^b=cg,d^b=dh,f^b=fh,g^b=gi,e^c=ehi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 245);