Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #384252 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14375 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
982 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13,8,12,6,27)(3,9,16,33,17,31,28,32)(4,10,22,37,20,36,21,39)(5,23,25,40,26,11,24,38)(14,29,18,48,42,56,46,34)(15,43,44,57,45,30,47,58)(19,50,54,62,52,35,53,61)(41,63,51,60,64,55,49,59);(1,3)(2,9)(4,18)(5,15)(6,28)(7,16)(8,17)(10,34)(11,30)(12,31)(13,33)(14,21)(19,51)(20,46)(22,42)(23,43)(24,47)(25,44)(26,45)(27,32)(29,37)(35,60)(36,48)(38,58)(39,56)(40,57)(41,53)(49,52)(50,59)(54,64)(55,61)(62,63);(1,4)(2,10,12,36)(3,14)(5,19)(6,22)(7,21)(8,20)(9,29,31,56)(11,35,23,50)(13,39,27,37)(15,41)(16,46)(17,42)(18,28)(24,54)(25,53)(26,52)(30,55,43,63)(32,48,33,34)(38,62,40,61)(44,49)(45,64)(47,51)(57,59,58,60);(1,5,8,26)(2,11,12,23)(3,15,17,45)(4,19,20,52)(6,25,7,24)(9,30,31,43)(10,35,36,50)(13,40,27,38)(14,41,42,64)(16,47,28,44)(18,49,46,51)(21,54,22,53)(29,55,56,63)(32,58,33,57)(34,59,48,60)(37,62,39,61)Polycyclic presentation:
2^9|a^2=h,c^2=ef,d^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,g^a=gi,c^b=cg,d^b=de,g^b=gi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,g^d=gi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291451);