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 | |
1883 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)(3,13)(4,18,21,9)(5,22,25,37)(6,26)(7,12)(8,16)(10,24,38,28)(11,29)(14,40,44,60)(15,31,45,56)(17,39)(19,49)(20,35,55,48)(23,41,46,36)(27,32)(30,53,33,43)(34,51)(42,58)(47,54)(50,61)(52,57)(59,63)(62,64);(1,3)(2,8)(4,14)(5,23)(6,27)(7,17)(9,33)(10,36)(11,39)(12,32)(13,26)(15,24)(16,29)(18,30)(19,50)(20,53)(21,44)(22,56)(25,46)(28,45)(31,37)(34,61)(35,40)(38,41)(42,51)(43,55)(47,62)(48,60)(49,59)(52,64)(54,63)(57,58);(1,4)(2,9)(3,14)(5,19)(6,20)(7,21)(8,30)(10,34)(11,35)(12,18)(13,40)(15,42)(16,43)(17,44)(22,57)(23,50)(24,51)(25,52)(26,48)(27,53)(28,54)(29,55)(31,58)(32,33)(36,62)(37,49)(38,47)(39,60)(41,61)(45,63)(46,64)(56,59);(1,5,7,25)(2,10,12,38)(3,15,17,45)(4,19,21,52)(6,28,29,24)(8,31,32,56)(9,34,18,47)(11,22,26,37)(13,41,39,36)(14,42,44,63)(16,46,27,23)(20,54,55,51)(30,58,33,59)(35,57,48,49)(40,61,60,62)(43,64,53,50)Polycyclic presentation:
2^9|a^2=h,d^2=i,e^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.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290347);