Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #164127 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×Q8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
807 conjugacy classes of subgroups; only the 256 normal subgroups are shown.
This group has 807 conjugacy classes of subgroups — too many to draw. Only its 256 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) = 512 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6)(7,8)(10,13,14,15)(17,18,21,23,22,19,20,24);(1,2,5,6)(3,7,4,8);(1,3,5,4)(2,7,6,8)(9,10)(11,15)(12,14)(13,16)(17,19,22,18)(20,23,21,24);(1,4,5,3)(2,7,6,8)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=eh,d^2=e,f^2=i,g^2=h,b^a=be,c^a=cf,f^a=fh,d^b=de,f^c=fi,g^c=ghi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 131801);