Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #253459 |
| 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 | C2×#121 | C23 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
729 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(2,8)(3,13)(5,22)(6,27)(9,32)(10,35)(11,39)(12,31)(14,40)(15,41)(16,49)(17,43)(18,24)(19,37)(20,29)(23,25)(26,52)(28,59)(30,34)(33,61)(36,48)(42,58)(44,51)(45,47)(46,63)(50,55)(53,60)(62,64);(1,2,7,12)(3,14,44,64)(4,8,21,31)(5,23,22,10)(6,28,29,59)(9,33,48,50)(11,27,39,20)(13,40,51,62)(15,16,30,58)(17,46,18,47)(19,35,37,25)(24,45,43,63)(26,57,53,56)(32,61,36,55)(34,42,41,49)(38,52,54,60);(1,3,7,18)(2,9,12,15)(4,17,21,44)(5,24,22,13)(6,16,29,33)(8,30,31,48)(10,36,23,34)(11,14,39,47)(19,51,37,43)(20,50,27,58)(25,41,35,32)(26,55,53,42)(28,46,59,64)(38,40,54,45)(49,60,61,52)(56,63,57,62);(1,2,7,12)(3,15,18,9)(4,8,21,31)(5,25,22,35)(6,11,29,39)(10,37,23,19)(13,41,24,32)(14,16,47,33)(17,48,44,30)(20,28,27,59)(26,56,53,57)(34,43,36,51)(38,60,54,52)(40,49,45,61)(42,63,55,62)(46,50,64,58)Polycyclic presentation:
2^9|b^2=ehi,c^2=i,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,f^b=fi,g^b=gh,d^c=di,e^c=eh,f^c=fi,f^d=fh44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 160677);