Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10792 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | C4×C22⋊C4 |
| commutator | C2×C4 | C43 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C2 | #10696 |
| prime radical | C24 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×M32 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
569 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,5,11,7,13,26,36,8,12,6,27,25,37,28,55)(3,14,17,39,19,40,47,61,16,9,18,30,45,31,48,59)(4,20,21,50,22,34,53,35,23,51,24,52,54,10,32,33)(15,41,43,62,44,29,56,57,42,49,46,63,64,38,60,58);(1,3)(2,9,12,14)(4,15)(5,17)(6,18)(7,19)(8,16)(10,29,34,38)(11,30,27,39)(13,31,37,40)(20,49,51,41)(21,43)(22,44)(23,42)(24,46)(25,45)(26,47)(28,48)(32,60)(33,57,35,58)(36,59,55,61)(50,63,52,62)(53,56)(54,64);(1,4,6,21,7,22,28,53,8,23,5,24,25,54,26,32)(2,10,11,35,13,20,36,52,12,34,27,33,37,51,55,50)(3,15,18,43,19,44,48,56,16,42,17,46,45,64,47,60)(9,29,30,58,31,49,59,62,14,38,39,57,40,41,61,63)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=g,e^2=h,g^2=h,h^2=i,b^a=bd,c^a=ch,f^a=fi,g^a=gi,d^b=di,e^c=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72319726, #72319734, #72319754, … |
| maximal normal subgroup of | #72319726, #72319734, #72319754, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53521);