Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468843 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16632 |
| 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 | |
1034 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,29,8,13,7,12)(3,14,17,9,18,34,30,33)(4,19,22,10,23,37,21,38)(5,24,26,11,28,41,27,40)(15,35,46,42,47,31,45,53)(16,48,50,56,51,32,52,43)(20,36,54,39,55,61,25,57)(44,62,63,58,64,59,49,60);(1,3)(2,9,13,33)(4,15)(5,16)(6,17)(7,30)(8,18)(10,35,38,31)(11,32,40,48)(12,14,29,34)(19,53,37,42)(20,44)(21,45)(22,46)(23,47)(24,56,41,43)(25,49)(26,50)(27,52)(28,51)(36,60,61,58)(39,62,57,59)(54,63)(55,64);(1,4)(2,10,13,38)(3,15)(5,25)(6,22)(7,21)(8,23)(9,31,33,35)(11,39,40,57)(12,19,29,37)(14,42,34,53)(16,49)(17,46)(18,47)(20,26)(24,36,41,61)(27,55)(28,54)(30,45)(32,59,48,62)(43,60,56,58)(44,50)(51,63)(52,64);(1,5,8,28)(2,11,13,40)(3,16,18,51)(4,20,23,55)(6,27,7,26)(9,32,33,48)(10,36,38,61)(12,41,29,24)(14,43,34,56)(15,44,47,64)(17,52,30,50)(19,39,37,57)(21,54,22,25)(31,58,35,60)(42,59,53,62)(45,63,46,49)Polycyclic presentation:
2^9|a^2=fg,b^2=g,c^2=g,d^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei,e^d=ei,f^d=fi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 376040);