Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #383548 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14361 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1333 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,14)(3,15,18,9,19,35,30,34)(4,20,23,38,25,10,24,39)(5,26,28,42,29,41,31,11)(16,43,21,32,46,36,45,60)(17,33,47,40,48,55,27,57)(22,51,53,49,54,56,52,37)(44,62,58,61,64,59,50,63);(1,3)(2,9)(4,21)(5,27)(6,30)(7,18)(8,19)(10,36)(11,40)(12,35)(13,34)(14,15)(16,24)(17,28)(20,43)(22,44)(23,46)(25,45)(26,55)(29,47)(31,48)(32,38)(33,41)(37,62)(39,60)(42,57)(49,59)(50,52)(51,61)(53,58)(54,64)(56,63);(1,4)(2,10)(3,16)(5,22)(6,23)(7,24)(8,25)(9,32)(11,37)(12,38)(13,20)(14,39)(15,36)(17,44)(18,45)(19,46)(21,30)(26,56)(27,58)(28,52)(29,54)(31,53)(33,59)(34,60)(35,43)(40,61)(41,51)(42,49)(47,50)(48,64)(55,62)(57,63);(1,5,6,28,8,29,7,31)(2,11,12,41,13,42,14,26)(3,17,18,27,19,48,30,47)(4,22,23,52,25,54,24,53)(9,33,15,40,34,55,35,57)(10,37,38,51,20,49,39,56)(16,44,45,58,46,64,21,50)(32,59,36,61,60,62,43,63)Polycyclic presentation:
2^9|a^2=h,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,g^c=gi,h^c=hi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290746);