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 | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
745 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(4,18)(5,22)(8,27)(9,30)(11,37)(13,38)(16,45)(20,52)(21,49)(23,46)(24,25)(28,31)(29,56)(32,34)(33,61)(35,39)(40,64)(41,63)(43,53)(48,54)(50,62)(51,55)(57,58)(59,60);(1,2,6,12)(3,13)(4,9,18,30)(5,11,24,39)(7,10,26,36)(8,23)(14,40)(15,38)(16,29)(17,41)(19,50)(20,33,48,59)(21,34,49,32)(22,37,25,35)(27,45)(28,55)(31,53)(42,64)(43,57)(44,63)(46,56)(47,62)(51,58)(52,61,54,60);(1,3,7,17)(2,8,10,27)(4,19,21,14)(5,23,25,45)(6,15,26,44)(9,31,34,28)(11,38,35,63)(12,29,36,56)(13,37,41,39)(16,22,46,24)(18,47,49,42)(20,53,52,55)(30,58,32,57)(33,62,61,64)(40,59,50,60)(43,48,51,54);(1,4,7,21)(2,9,10,34)(3,14,17,19)(5,20,25,52)(6,18,26,49)(8,28,27,31)(11,33,35,61)(12,30,36,32)(13,40,41,50)(15,42,44,47)(16,43,46,51)(22,54,24,48)(23,55,45,53)(29,57,56,58)(37,60,39,59)(38,64,63,62)Polycyclic presentation:
2^9|b^2=eh,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, 161745);