Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #94427 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1390 |
| commutator | C22×Q8 | C42 |
| Frattini, cosocle | C22×#124 | C22 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C22×Q8 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C22×#124 ◁ G | |
| LECS | G ▷ C22×#124 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
495 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 495 conjugacy classes of subgroups — too many to draw. Only its 48 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)(3,7)(4,6)(5,8)(9,10,12,15)(11,18,20,24)(13,16,22,23)(14,17,21,19);(1,3)(2,5,6,7)(4,8)(9,10,13,16,14,17,20,24)(11,19,21,23,22,15,12,18)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=i,e^2=i,f^2=i,b^a=bc,c^a=cf,e^a=egi,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=df,g^b=gi,h^b=hi,d^c=dhi,e^c=ei,f^c=fi,e^d=ei,f^d=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1624);