Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #93550 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1295 |
| commutator | C2×C42 | C42 |
| Frattini, cosocle | C22×#97 | C22 |
| prime radical | C24 | #100 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
304 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 304 conjugacy classes of subgroups — too many to draw. Only its 62 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:
(1,2,4,7,5,8,3,6)(9,10,11,17,12,15,21,22,14,18,20,24,13,16,19,23);(2,6,8,7)(10,15)(11,19)(12,13)(16,18)(17,24)(20,21)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fh,d^2=f,f^2=i,h^2=i,b^a=bc,c^a=cf,e^a=efgh,g^a=gi,h^a=hi,c^b=cg,d^b=dh,f^b=fi,h^b=hi,d^c=di56 conjugacy classes — view the full character table.
| maximal subgroup of | #72280100 |
| maximal normal subgroup of | #72280100 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 746);