Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #360226 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2352 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #22750 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
521 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,14,9,13,25,42,6,15,26,11,7,32,4,23)(3,16,20,43,21,45,17,46,22,10,33,34,19,36,48,52)(5,12,29,41,30,40,55,62,31,39,54,37,28,57,24,53)(18,44,50,64,51,61,47,63,49,35,58,59,27,38,56,60);(1,3,7,19,6,22,9,21)(2,10,13,36,15,16,32,45)(4,17,26,20,25,48,8,33)(5,27,28,49,31,51,30,18)(11,34,23,52,14,43,42,46)(12,38,40,44,39,61,57,35)(24,50,54,56,55,58,29,47)(37,60,53,64,41,63,62,59);(1,4,7,26,6,25,9,8)(2,11,13,23,15,14,32,42)(3,17,19,20,22,48,21,33)(5,24,28,54,31,55,30,29)(10,34,36,52,16,43,45,46)(12,37,40,53,39,41,57,62)(18,47,27,50,49,56,51,58)(35,59,38,60,44,64,61,63);(1,5)(2,12)(3,18)(4,24)(6,31)(7,28)(8,29)(9,30)(10,35)(11,37)(13,40)(14,41)(15,39)(16,44)(17,47)(19,27)(20,50)(21,51)(22,49)(23,53)(25,55)(26,54)(32,57)(33,58)(34,59)(36,38)(42,62)(43,64)(45,61)(46,63)(48,56)(52,60)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=f,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,d^b=deh,e^b=ei,g^b=gi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405175);