Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #250455 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #18240 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
966 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)(3,14)(4,18)(5,23)(9,31)(11,19)(12,42)(13,30)(15,43)(16,47)(17,21)(20,27)(22,52)(24,25)(28,34)(29,35)(32,60)(33,40)(37,62)(41,50)(44,54)(45,53)(46,51)(48,64)(49,55)(56,63)(57,58)(59,61);(1,2,7,13)(3,10,15,39)(4,19,20,50)(5,9,24,12)(6,11,26,41)(8,27,30,18)(14,29,43,58)(16,32,45,37)(17,38,46,36)(21,35,51,57)(22,42,52,31)(23,40,25,34)(28,49,33,55)(44,63,64,56)(47,59,53,61)(48,62,54,60);(2,9)(4,20)(5,24)(8,28)(10,32)(11,34)(12,13)(14,43)(16,45)(18,27)(19,31)(21,51)(23,25)(29,56)(30,33)(35,59)(36,60)(37,39)(38,62)(40,41)(42,50)(48,54)(57,61)(58,63);(1,3,6,17)(2,10,11,38)(4,21,18,14)(5,16,25,48)(7,15,26,46)(8,29,19,35)(9,32,34,62)(12,37,40,60)(13,39,41,36)(20,51,27,43)(22,53,49,44)(23,54,24,45)(28,56,31,59)(30,58,50,57)(33,63,42,61)(47,55,64,52)Polycyclic presentation:
2^9|b^2=i,d^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=ghi,c^b=cg,e^b=eh,f^b=fh,g^b=gi,e^c=ei,g^c=gi,e^d=eh,f^d=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 157668);