Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23387 |
| 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×#376 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
810 conjugacy classes of subgroups; only the 298 normal subgroups are shown.
This group has 810 conjugacy classes of subgroups — too many to draw. Only its 298 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2)(3,6)(4,5)(7,8)(17,18,22,26,23,27,21,37)(19,28,32,39,30,42,31,24)(20,33,35,40,36,47,34,25)(29,43,46,48,44,38,45,41);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,21,31,23,30,22,32)(18,24,26,28,27,39,37,42)(20,29,34,45,36,44,35,46)(25,38,33,41,40,43,47,48);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,20,21,34,23,36,22,35)(18,25,26,33,27,40,37,47)(19,29,31,45,30,44,32,46)(24,38,28,41,39,43,42,48);(9,11,12,15)(10,14,13,16)(19,30)(24,39)(28,42)(29,44)(31,32)(38,43)(41,48)(45,46)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=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266158);