Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #383150 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14357 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60368 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1899 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,28)(3,14,17,9,18,34,29,33)(4,19,21,37,22,10,30,38)(5,23,26,11,27,40,25,41)(15,42,47,35,46,59,45,31)(16,32,24,43,48,56,49,39)(20,51,53,36,54,57,55,50)(44,58,64,62,63,60,52,61);(1,3)(2,9)(4,15)(5,24)(6,29)(7,17)(8,18)(10,35)(11,39)(12,34)(13,33)(14,28)(16,25)(19,31)(20,52)(21,47)(22,46)(23,56)(26,48)(27,49)(30,45)(32,40)(36,61)(37,42)(38,59)(41,43)(44,53)(50,62)(51,60)(54,64)(55,63)(57,58);(1,4)(2,10)(3,15)(5,20)(6,21)(7,30)(8,22)(9,31)(11,36)(12,37)(13,19)(14,42)(16,44)(17,45)(18,46)(23,57)(24,52)(25,53)(26,55)(27,54)(28,38)(29,47)(32,58)(33,35)(34,59)(39,62)(40,51)(41,50)(43,61)(48,63)(49,64)(56,60);(1,5,8,27)(2,11,13,41)(3,16,18,48)(4,20,22,54)(6,26,7,25)(9,32,33,56)(10,36,19,50)(12,23,28,40)(14,43,34,39)(15,44,46,63)(17,49,29,24)(21,55,30,53)(31,58,35,60)(37,57,38,51)(42,61,59,62)(45,64,47,52)Polycyclic presentation:
2^9|a^2=h,d^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^c=hi,e^d=ei,h^d=hi65 conjugacy classes — view the full character table.
| maximal quotient of | #72239913 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290349);