Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7402 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #353 |
| commutator | C2×C42 | C42 |
| Frattini, cosocle | #2297 | C22 |
| socle | C22 | #1393 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C23×C4 ◁ #2297 ◁ G | |
| LECS | G ▷ #2297 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
403 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 403 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)(3,5)(4,7)(6,8)(9,10,12,16,14,13,23,17)(11,18,20,21,22,24,19,15);(2,5,7,6)(9,11,13,21)(10,15,14,22)(12,20,17,24)(16,18,23,19)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=f,d^2=i,g^2=i,b^a=bc,c^a=cf,e^a=efg,g^a=gi,c^b=cg,f^b=fi,g^b=gh,h^b=hi,e^c=ehi,g^e=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1473);