Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11234 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #391 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C2 | #10877 |
| prime radical | C24 | #131 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
493 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,5,11,6,12,23,36,7,10,24,33,25,34,51,60)(3,13,16,35,17,38,42,62,15,8,22,27,41,28,50,53)(4,9,20,30,21,32,48,58,19,31,45,57,46,59,63,64)(14,26,40,52,18,29,44,55,39,37,49,61,43,54,47,56);(1,3)(2,8)(4,18,19,43)(5,22)(6,17)(7,15)(9,29,31,54)(10,13)(11,35)(12,28)(14,21,39,46)(16,24)(20,47,45,44)(23,50)(25,41)(26,32,37,59)(27,33)(30,56,57,55)(34,38)(36,62)(40,63,49,48)(42,51)(52,64,61,58)(53,60);(1,4,5,20,6,21,23,48,7,19,24,45,25,46,51,63)(2,9,11,30,12,32,36,58,10,31,33,57,34,59,60,64)(3,14,16,40,17,18,42,44,15,39,22,49,41,43,50,47)(8,26,27,52,28,29,53,55,13,37,35,61,38,54,62,56)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=ch,e^b=ei,f^b=fi,d^c=di,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72239196, #72239197, #72239200, … |
| maximal normal subgroup of | #72239196, #72239197, #72239200, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54361);