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 | |
731 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(2,9)(3,15)(5,24)(6,23)(10,36)(11,27)(13,14)(16,46)(17,49)(18,26)(19,50)(20,45)(21,52)(22,29)(25,34)(28,60)(31,32)(33,38)(37,63)(39,40)(41,57)(43,47)(44,54)(48,64)(51,58)(53,62)(55,56)(59,61);(1,2,8,14)(3,16,21,53)(4,9,7,13)(5,25,24,11)(6,30,32,35)(10,37,33,51)(12,23,42,31)(15,43,52,64)(17,19,39,54)(18,47,20,48)(22,27,29,34)(26,46,45,62)(28,61,56,59)(36,50,38,44)(40,58,49,63)(41,60,57,55);(1,3,8,18)(2,10,14,17)(4,21,7,20)(5,26,24,15)(6,19,32,37)(9,33,13,39)(11,40,25,38)(12,16,42,47)(22,45,29,52)(23,54,31,51)(27,36,34,49)(28,58,56,44)(30,48,35,53)(41,43,57,46)(50,55,63,60)(59,62,61,64);(1,2,8,14)(3,17,18,10)(4,9,7,13)(5,27,24,34)(6,12,32,42)(11,22,25,29)(15,36,26,49)(16,19,47,37)(20,33,21,39)(23,35,31,30)(28,59,56,61)(38,45,40,52)(41,55,57,60)(43,50,46,63)(44,62,58,64)(48,51,53,54)Polycyclic presentation:
2^9|b^2=eh,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, 160665);