Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #94427 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1390 |
| commutator | C22×Q8 | C42 |
| Frattini, cosocle | C22×C4⋊Q8 | C22 |
| prime radical | C24 | #99 |
| LCS | G ▷ C22×Q8 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C22×C4⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4⋊Q8 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
445 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 445 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)(9,10,13,18,15,11,20,17)(12,22,14,21,23,24,16,19);(1,3)(2,5,6,7)(4,8)(9,11,14,21,15,10,16,19)(12,18,20,24,23,17,13,22)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=i,d^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, 1626);