Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #457619 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16298 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1463 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,14,8,12,7,13)(3,15,18,33,19,9,28,38)(4,20,23,10,24,39,29,35)(5,25,26,11,27,40,30,37)(16,43,45,41,21,52,46,31)(17,47,48,32,50,57,49,42)(22,34,54,58,55,36,56,51)(44,59,62,61,53,60,63,64);(1,3)(2,9)(4,21)(5,17)(6,28)(7,18)(8,19)(10,31)(11,32)(12,15)(13,38)(14,33)(16,24)(20,43)(22,53)(23,45)(25,57)(26,49)(27,50)(29,46)(30,48)(34,59)(35,41)(36,60)(37,42)(39,52)(40,47)(44,55)(51,61)(54,62)(56,63)(58,64);(1,4,6,23,8,24,7,29)(2,10,13,20,12,35,14,39)(3,16,18,45,19,21,28,46)(5,22,26,54,27,55,30,56)(9,31,33,52,15,41,38,43)(11,36,25,58,37,34,40,51)(17,44,48,62,50,53,49,63)(32,60,47,61,42,59,57,64);(1,5,8,27)(2,11,12,37)(3,17,19,50)(4,22,24,55)(6,26,7,30)(9,32,15,42)(10,34,35,36)(13,25,14,40)(16,44,21,53)(18,48,28,49)(20,51,39,58)(23,54,29,56)(31,59,41,60)(33,47,38,57)(43,61,52,64)(45,62,46,63)Polycyclic presentation:
2^9|a^2=ef,c^2=f,d^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,f^b=fi,d^c=de,g^c=gi,h^c=hi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 364816);