Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1406 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #353 |
| commutator | C42 | C4×C8 |
| Frattini, cosocle | C4×#124 | C22 |
| socle | C22 | #1393 |
| prime radical | C22×D4 | C4⋊C4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ C4×#124 ◁ G | |
| LECS | G ▷ C4×#124 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
243 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 243 conjugacy classes of subgroups — too many to draw. Only its 56 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:
(1,2,4,7,6,8,5,3)(9,10,11,17)(12,18,22,15)(13,16,21,23)(14,19,20,24);(1,3,5,8,6,7,4,2)(9,11,12,21,14,20,13,22)(10,15,18,17,19,23,16,24)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=h,d^2=g,e^2=gi,f^2=i,b^a=bc,c^a=cf,e^a=ehi,f^a=fh,h^a=hi,c^b=ci,d^b=dfhi,d^c=dhi,e^d=ei,f^d=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1355);