Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #478423 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17147 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
942 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,29,8,12,7,13)(3,14,19,40,17,9,18,33)(4,20,23,36,24,10,30,37)(5,25,26,11,28,41,27,39)(15,42,50,51,44,31,21,34)(16,45,49,32,47,58,48,38)(22,54,55,35,56,59,57,52)(43,63,53,61,46,62,64,60);(1,3)(2,9,12,14)(4,21)(5,16)(6,19)(7,18)(8,17)(10,34,20,51)(11,38,39,32)(13,40,29,33)(15,23)(22,53)(24,50)(25,58,41,45)(26,49)(27,48)(28,47)(30,44)(31,37,42,36)(35,63,52,62)(43,57)(46,55)(54,60,59,61)(56,64);(1,4)(2,10)(3,15)(5,22)(6,30)(7,23)(8,24)(9,31)(11,35)(12,20)(13,37)(14,42)(16,46)(17,44)(18,50)(19,21)(25,59)(26,57)(27,55)(28,56)(29,36)(32,61)(33,51)(34,40)(38,60)(39,52)(41,54)(43,47)(45,62)(48,64)(49,53)(58,63);(1,5,8,28)(2,11,12,39)(3,16,17,47)(4,22,24,56)(6,27,7,26)(9,32,14,38)(10,35,20,52)(13,41,29,25)(15,43,44,46)(18,49,19,48)(21,53,50,64)(23,57,30,55)(31,60,42,61)(33,45,40,58)(34,62,51,63)(36,59,37,54)Polycyclic presentation:
2^9|a^2=fg,b^2=ef,d^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=def,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,g^c=gi,g^d=gi,h^d=hi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 385620);