Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #93165 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1304 |
| commutator | C23×C4 | C42 |
| Frattini, cosocle | C22×C4⋊Q8 | C22 |
| prime radical | C24 | #95 |
| LCS | G ▷ C23×C4 ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C4⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4⋊Q8 ▷ C24 ▷ C22 ▷ C1 | |
471 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 471 conjugacy classes of subgroups — too many to draw. Only its 52 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)(3,7,8,5)(4,6)(9,10,13,18,15,11,20,22)(12,21,14,19,24,17,16,23);(2,5)(6,7)(9,11,14,23,15,10,16,19)(12,17,20,18,24,21,13,22)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=i,d^2=i,e^2=i,b^a=bc,c^a=cf,e^a=egi,f^a=fh,g^a=gi,c^b=cg,d^b=dfi,f^b=fi,g^b=gi,d^c=dh,e^c=ei,e^d=ei44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 371);