Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #148578 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #11196 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C22×2−1+4 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1281 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)(3,13,17,34)(4,18)(5,10)(6,26,27,11)(7,12)(8,15,30,23)(9,21,33,53)(14,39,19,48)(16,43)(20,46)(22,47)(24,55,56,35)(25,36)(28,58)(29,42)(31,60)(32,51,62,64)(37,50)(38,44)(40,45,49,59)(41,61)(52,63)(54,57);(1,3,7,17)(2,8,12,30)(4,19,22,14)(5,23,25,15)(6,16,27,44)(9,31,33,28)(10,34,36,13)(11,29,26,57)(18,45,47,59)(20,40,52,49)(21,50,53,41)(24,54,56,42)(32,37,62,61)(35,43,55,38)(39,63,48,46)(51,58,64,60);(1,4,5,20,7,22,25,52)(2,9,10,32,12,33,36,62)(3,14,15,40,17,19,23,49)(6,21,24,51,27,53,56,64)(8,28,13,37,30,31,34,61)(11,18,35,46,26,47,55,63)(16,41,42,58,44,50,54,60)(29,59,38,39,57,45,43,48)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=d,d^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,c^b=cf,d^b=di,g^b=gi,f^c=fi,g^c=gh,g^e=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72267966 |
| maximal normal subgroup of | #72267966 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55775);