Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #402001 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14828 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
752 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,27,7,12,6,13)(3,14,17,39,18,9,28,32)(4,19,26,10,22,36,21,34)(5,23,29,35,25,11,24,53)(15,40,42,30,43,47,46,37)(16,31,44,57,45,38,55,59)(20,49,54,52,51,48,50,33)(41,61,62,58,63,60,64,56);(1,3,6,17,7,18,8,28)(2,9,13,32,12,14,27,39)(4,15,21,42,22,43,26,46)(5,16,24,44,25,45,29,55)(10,30,19,47,34,37,36,40)(11,31,35,57,23,38,53,59)(20,41,50,62,51,63,54,64)(33,56,48,61,52,58,49,60);(1,4,7,22)(2,10,12,34)(3,15,18,43)(5,20,25,51)(6,26,8,21)(9,30,14,37)(11,33,23,52)(13,36,27,19)(16,41,45,63)(17,46,28,42)(24,54,29,50)(31,56,38,58)(32,40,39,47)(35,49,53,48)(44,64,55,62)(57,60,59,61);(1,5)(2,11,12,23)(3,16)(4,20)(6,24)(7,25)(8,29)(9,31,14,38)(10,33,34,52)(13,35,27,53)(15,41)(17,44)(18,45)(19,48,36,49)(21,50)(22,51)(26,54)(28,55)(30,56,37,58)(32,57,39,59)(40,60,47,61)(42,62)(43,63)(46,64)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=i,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^b=gi,f^c=fi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 309198);