Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13353 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2021 |
| commutator | C2×#41 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| socle | C2 | #11069 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1332 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,7,11,8,13,5,25)(3,14,18,32,19,41,16,9)(4,20,23,50,24,51,21,52)(6,12,29,36,28,38,26,56)(10,33,34,54,35,53,55,22)(15,42,45,61,46,63,43,49)(17,40,48,37,27,57,47,31)(30,44,39,60,59,58,62,64);(1,3,5,16,8,19,7,18)(2,9,11,14,13,32,25,41)(4,15,21,43,24,46,23,45)(6,27,26,48,28,17,29,47)(10,30,34,39,35,59,55,62)(12,37,36,57,38,31,56,40)(20,49,50,42,51,61,52,63)(22,44,53,64,54,58,33,60);(1,4)(2,10)(3,15)(5,21)(6,22)(7,23)(8,24)(9,30)(11,34)(12,20)(13,35)(14,39)(16,43)(17,44)(18,45)(19,46)(25,55)(26,53)(27,58)(28,54)(29,33)(31,49)(32,59)(36,50)(37,61)(38,51)(40,42)(41,62)(47,64)(48,60)(52,56)(57,63)Polycyclic presentation:
2^9|a^2=g,b^2=d,d^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fhi,c^b=cf,e^b=eh,h^c=hi,f^e=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72342089, #72342095 |
| maximal normal subgroup of | #72342095 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55373);