Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #254539 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
715 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13)(3,15,18,45)(4,6,24,10)(5,11,28,41)(8,14,31,23)(9,32,35,19)(12,30,42,38)(16,48,29,43)(17,44,33,52)(20,47,51,64)(21,54,55,49)(22,27,26,37)(25,39,56,57)(34,50,61,46)(36,62,58,53)(40,60,59,63);(3,16,18,29)(5,25)(9,33,35,17)(11,39)(15,43,45,48)(19,52,32,44)(20,34,51,61)(21,36,55,58)(22,40)(26,59)(27,60)(28,56)(37,63)(41,57)(46,64,50,47)(49,62,54,53);(1,3,8,20)(2,9,14,36)(4,21,12,17)(5,19,25,53)(6,29,30,61)(7,18,31,51)(10,16,38,34)(11,15,39,47)(13,35,23,58)(22,48,40,50)(24,55,42,33)(26,43,59,46)(27,54,60,44)(28,32,56,62)(37,49,63,52)(41,45,57,64);(1,4,8,12)(2,10,14,38)(3,17,20,21)(5,26,25,59)(6,23,30,13)(7,24,31,42)(9,34,36,16)(11,27,39,60)(15,44,47,54)(18,33,51,55)(19,46,53,43)(22,56,40,28)(29,35,61,58)(32,50,62,48)(37,57,63,41)(45,52,64,49)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=i,d^2=i,g^2=h,b^a=be,c^a=cf,d^a=deh,g^a=ghi,c^b=cg,f^b=fi,d^c=di,e^c=eh,g^c=gh,f^d=fh44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 161751);