Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #469731 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16661 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
899 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)(3,9)(4,19)(5,25,27,56)(6,30)(7,12)(8,13)(10,22)(11,29,38,28)(14,40)(15,44,46,54)(16,39)(17,33)(18,34)(20,49)(21,37,26,35)(23,51)(24,36)(31,42)(32,48,60,47)(41,59,45,57)(43,58)(50,53,55,52)(61,63,64,62);(1,3,8,18)(2,9,13,34)(4,20,23,14)(5,15,27,46)(6,17,7,16)(10,31,36,58)(11,32,38,60)(12,39,30,33)(19,49,51,40)(21,45,26,41)(22,42,24,43)(25,54,56,44)(28,48,29,47)(35,57,37,59)(50,64,55,61)(52,62,53,63);(1,4,6,22,8,23,7,24)(2,10,12,19,13,36,30,51)(3,14,16,42,18,20,17,43)(5,26,28,53,27,21,29,52)(9,31,33,40,34,58,39,49)(11,37,25,55,38,35,56,50)(15,45,47,63,46,41,48,62)(32,59,44,64,60,57,54,61);(1,5)(2,11,13,38)(3,15)(4,21,23,26)(6,28)(7,29)(8,27)(9,32,34,60)(10,35)(12,25,30,56)(14,41,20,45)(16,47)(17,48)(18,46)(19,50)(22,52,24,53)(31,57)(33,44,39,54)(36,37)(40,61)(42,62,43,63)(49,64)(51,55)(58,59)Polycyclic presentation:
2^9|a^2=e,b^2=i,c^2=f,d^2=g,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,h^b=hi,d^c=de,g^c=gi,e^d=ei53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 376928);