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 | #18242 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
947 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)(4,19)(5,24)(9,31)(11,20)(12,42)(13,30)(16,43)(17,47)(18,22)(21,27)(23,52)(25,26)(28,34)(29,36)(32,60)(33,41)(35,50)(38,62)(44,54)(45,51)(46,53)(48,64)(49,55)(56,63)(57,58)(59,61);(1,2,7,13)(3,10,16,40)(4,20,21,50)(5,9,25,12)(6,11,15,35)(8,27,30,19)(14,29,43,58)(17,32,46,38)(18,39,45,37)(22,36,51,57)(23,42,52,31)(24,41,26,34)(28,49,33,55)(44,63,64,56)(47,59,53,61)(48,62,54,60);(2,9)(4,21)(5,25)(8,28)(10,32)(11,34)(12,13)(14,43)(17,46)(19,27)(20,31)(22,51)(24,26)(29,56)(30,33)(35,41)(36,59)(37,60)(38,40)(39,62)(42,50)(48,54)(57,61)(58,63);(1,3,15,45)(2,10,35,37)(4,22,27,43)(5,17,24,54)(6,18,7,16)(8,29,50,57)(9,32,41,60)(11,39,13,40)(12,38,34,62)(14,21,51,19)(20,36,30,58)(23,53,55,64)(25,46,26,48)(28,56,42,61)(31,59,33,63)(44,52,47,49)Polycyclic presentation:
2^9|b^2=i,d^2=hi,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, 157670);