Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32712 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1057 conjugacy classes of subgroups; only the 242 normal subgroups are shown.
This group has 1057 conjugacy classes of subgroups — too many to draw. Only its 242 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,5)(3,6,7,8)(9,10,12,13)(11,15,16,14)(17,18)(19,22)(20,24)(21,23)(25,29)(26,30)(27,32)(28,31);(2,5)(3,7)(10,13)(14,15)(19,25,26,27)(22,29,30,32);(3,7)(6,8)(9,11)(10,14)(12,16)(13,15)(17,19)(18,22)(20,27)(21,26)(23,30)(24,32)(25,28)(29,31);(1,3,4,7)(2,6,5,8)(11,16)(14,15)(18,23)(22,30)(24,31)(29,32)Polycyclic presentation:
2^9|a^2=e,b^2=h,d^2=ef,g^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=def,d^c=de,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299130);