Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23391 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
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×C4⋊C4 |
| prime radical | C24 | C22×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
862 conjugacy classes of subgroups; only the 306 normal subgroups are shown.
This group has 862 conjugacy classes of subgroups — too many to draw. Only its 306 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) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(17,18,21,27,23,28,22,26)(19,25,20,24,29,32,30,31);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,30,23,29,21,20)(18,24,27,32,28,31,26,25);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,20,21,29,23,30,22,19)(18,25,26,31,28,32,27,24);(9,11,12,15)(10,14,13,16)(19,29)(20,30)(24,31)(25,32)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=e,d^2=g,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,c^b=cg,d^b=di,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266159);