Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31418 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2688 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14392 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
819 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)(3,9,18,32,17,33,16,45)(4,19,23,10,22,36,29,35)(5,24,27,37,26,53,25,11)(14,38,40,30,20,48,46,56)(15,41,43,57,44,63,42,31)(21,34,51,52,50,59,54,47)(39,55,61,62,49,58,64,60);(1,3)(2,9)(4,20,22,14)(5,15)(6,16)(7,18)(8,17)(10,30,35,56)(11,31)(12,32)(13,33)(19,38,36,48)(21,39,50,49)(23,46,29,40)(24,41)(25,42)(26,44)(27,43)(28,45)(34,58,59,55)(37,57)(47,62,52,60)(51,61,54,64)(53,63);(1,4)(2,10)(3,14)(5,21)(6,29)(7,23)(8,22)(9,30)(11,34)(12,36)(13,35)(15,39)(16,46)(17,20)(18,40)(19,28)(24,52)(25,54)(26,50)(27,51)(31,55)(32,48)(33,56)(37,59)(38,45)(41,62)(42,64)(43,61)(44,49)(47,53)(57,58)(60,63);(1,5,6,25,8,26,7,27)(2,11,12,24,13,37,28,53)(3,15,16,42,17,44,18,43)(4,21,29,54,22,50,23,51)(9,31,32,41,33,57,45,63)(10,34,36,52,35,59,19,47)(14,39,46,64,20,49,40,61)(30,55,48,62,56,58,38,60)Polycyclic presentation:
2^9|a^2=h,b^2=e,d^2=ef,f^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^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 292093);