Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1355 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #329 |
| commutator | C42 | C4×C8 |
| Frattini, cosocle | C4×#97 | C22 |
| socle | C22 | #1295 |
| prime radical | C23 | #343 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C4×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
205 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 205 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) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2,3,5,4,6,7,8)(10,14)(11,18)(12,20)(15,23)(16,17)(19,21);(9,10,11,15,12,17,18,24,13,16,19,23,20,14,21,22)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=f,d^2=h,e^2=g,g^2=i,b^a=bc,c^a=cf,e^a=efg,g^a=gi,c^b=cg,f^b=fi,e^c=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 636);