Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #376489 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14181 |
| 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 | |
1072 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)(3,8,15,30)(4,17)(5,10,22,35)(6,24)(7,11)(9,20)(12,26)(13,37,18,46)(14,40)(16,44,43,29)(19,47,49,58)(21,36)(23,52,51,34)(25,33)(27,45,31,39)(28,42)(32,53,57,50)(38,60)(41,55)(48,59)(54,63)(56,61)(62,64);(1,3,6,16,7,15,26,43)(2,8,12,29,11,30,24,44)(4,18,20,45,21,13,25,39)(5,14,23,41,22,42,51,64)(9,31,17,46,33,27,36,37)(10,28,34,55,35,40,52,62)(19,48,50,63,49,38,53,61)(32,56,47,60,57,54,58,59);(1,4,7,21)(2,9,11,33)(3,13,15,18)(5,19,22,49)(6,25,26,20)(8,27,30,31)(10,32,35,57)(12,36,24,17)(14,38,42,48)(16,45,43,39)(23,53,51,50)(28,54,40,56)(29,46,44,37)(34,58,52,47)(41,63,64,61)(55,60,62,59);(1,5)(2,10,11,35)(3,14)(4,19)(6,23)(7,22)(8,28,30,40)(9,32,33,57)(12,34,24,52)(13,38)(15,42)(16,41)(17,47,36,58)(18,48)(20,50)(21,49)(25,53)(26,51)(27,54,31,56)(29,55,44,62)(37,59,46,60)(39,61)(43,64)(45,63)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=i,d^2=eh,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,f^c=fi,e^d=ei,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 283686);