Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453836 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16146 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
626 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,14,6,13)(3,15,19,35,20,34,18,9)(4,21,24,39,26,38,25,10)(5,27,31,42,30,41,29,11)(16,43,22,36,46,32,45,59)(17,33,47,40,48,55,28,57)(23,37,52,49,54,62,53,56)(44,60,50,58,51,64,63,61);(1,3)(2,9,14,35)(4,22)(5,28,30,47)(6,18)(7,19)(8,20)(10,36,39,59)(11,40)(12,15,13,34)(16,25)(17,29,48,31)(21,32,38,43)(23,51,54,44)(24,46)(26,45)(27,55)(33,41)(37,61)(42,57)(49,60)(50,52,63,53)(56,64)(58,62);(1,4,6,24,8,26,7,25)(2,10,12,38,14,39,13,21)(3,16,18,22,20,46,19,45)(5,23,29,52,30,54,31,53)(9,32,15,36,35,43,34,59)(11,37,27,56,42,62,41,49)(17,44,28,50,48,51,47,63)(33,58,40,60,55,61,57,64);(1,5,7,31,8,30,6,29)(2,11,12,27,14,42,13,41)(3,17,18,28,20,48,19,47)(4,23,25,53,26,54,24,52)(9,33,34,57,35,55,15,40)(10,37,38,56,39,62,21,49)(16,44,22,50,46,51,45,63)(32,58,59,64,43,61,36,60)Polycyclic presentation:
2^9|a^2=efg,b^2=fh,c^2=e,d^2=fi,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361033);