Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #391560 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14641 |
| 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 | |
779 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)(4,17,21,9)(5,10)(6,12)(7,11)(13,42,44,29)(14,30)(15,32)(16,31)(18,48,47,33)(19,36,23,34)(20,41,27,35)(22,52)(24,37)(25,38)(26,40)(28,39)(43,57,45,56)(46,58)(49,61,50,59)(51,62,54,60)(53,64)(55,63);(1,3,7,16)(2,8,11,31)(4,18,21,47)(5,22,24,14)(6,26,28,15)(9,33,17,48)(10,30,37,52)(12,40,39,32)(13,27,44,20)(19,50,23,49)(25,46,55,53)(29,41,42,35)(34,59,36,61)(38,64,63,58)(43,51,45,54)(56,62,57,60);(1,4)(2,9)(3,13)(5,23)(6,27)(7,21)(8,29)(10,36)(11,17)(12,41)(14,45)(15,47)(16,44)(18,26)(19,24)(20,28)(22,43)(25,51)(30,57)(31,42)(32,48)(33,40)(34,37)(35,39)(38,60)(46,49)(50,53)(52,56)(54,55)(58,59)(61,64)(62,63);(1,5,6,25,7,24,28,55)(2,10,12,38,11,37,39,63)(3,14,15,46,16,22,26,53)(4,19,20,51,21,23,27,54)(8,30,32,58,31,52,40,64)(9,34,35,60,17,36,41,62)(13,43,18,49,44,45,47,50)(29,56,33,59,42,57,48,61)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=g,g^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,d^b=def,g^b=gi,d^c=de,g^c=gi,h^c=hi,e^d=ei53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 298757);