Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #473877 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16885 |
| 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 | |
849 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,7,11,27,29)(3,8,28,56,15,33,16,32)(4,17,20,36,21,40,26,9)(5,22,25,38,24,39,23,10)(13,41,49,30,18,50,43,34)(14,44,47,31,46,58,48,59)(19,35,54,55,53,37,52,51)(42,57,62,61,45,60,63,64);(1,3,6,16,7,15,27,28)(2,8,29,56,11,33,12,32)(4,18,20,49,21,13,26,43)(5,14,23,47,24,46,25,48)(9,34,40,41,36,30,17,50)(10,31,22,44,38,59,39,58)(19,45,52,63,53,42,54,62)(35,60,55,61,37,57,51,64);(1,4,7,21)(2,9,11,36)(3,13,15,18)(5,19,24,53)(6,26,27,20)(8,30,33,34)(10,37,38,35)(12,40,29,17)(14,45,46,42)(16,49,28,43)(22,55,39,51)(23,54,25,52)(31,57,59,60)(32,50,56,41)(44,61,58,64)(47,62,48,63);(1,5)(2,10)(3,14)(4,19)(6,23)(7,24)(8,31)(9,35)(11,38)(12,39)(13,42)(15,46)(16,47)(17,51)(18,45)(20,52)(21,53)(22,29)(25,27)(26,54)(28,48)(30,57)(32,58)(33,59)(34,60)(36,37)(40,55)(41,61)(43,62)(44,56)(49,63)(50,64)Polycyclic presentation:
2^9|a^2=fg,b^2=egh,c^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,g^b=gi,d^c=de,f^c=fi,h^c=hi,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 381074);