Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6352469 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2583 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,10,31)(3,16,34,57)(4,12,30,7)(5,13,35,56)(6,8,15,9)(11,36,51,19)(14,33,24,23)(17,44,58,63)(18,32,39,37)(20,45,21,43)(22,40,54,26)(25,41,52,53)(27,42,28,29)(38,59,64,47)(46,48,62,49)(50,55,61,60);(1,3,8,21)(2,11,15,39)(4,16,14,43)(5,17,27,49)(6,18,31,51)(7,32,23,19)(9,20,10,34)(12,37,33,36)(13,38,42,61)(22,44,25,46)(24,45,30,57)(26,55,53,47)(28,48,35,58)(29,50,56,64)(40,60,41,59)(52,62,54,63);(1,4,8,14)(2,12,15,33)(3,16,21,43)(5,25,27,22)(6,23,31,7)(9,24,10,30)(11,37,39,36)(13,41,42,40)(17,46,49,44)(18,19,51,32)(20,45,34,57)(26,56,53,29)(28,54,35,52)(38,59,61,60)(47,50,55,64)(48,63,58,62);(1,5,9,28)(2,13,6,29)(3,17,20,48)(4,22,24,52)(7,26,33,41)(8,27,10,35)(11,38,18,50)(12,40,23,53)(14,25,30,54)(15,42,31,56)(16,44,45,62)(19,47,37,60)(21,49,34,58)(32,55,36,59)(39,61,51,64)(43,46,57,63);(1,6,9,2)(3,18,20,11)(4,23,24,12)(5,13,28,29)(7,30,33,14)(8,31,10,15)(16,19,45,37)(17,38,48,50)(21,51,34,39)(22,40,52,53)(25,41,54,26)(27,42,35,56)(32,57,36,43)(44,60,62,47)(46,59,63,55)(49,61,58,64)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=h,d^2=i,e^2=i,f^2=h,b^a=bf,c^a=cg,e^a=eh,f^a=fh,f^b=fh,g^b=gi,d^c=dh,f^c=fi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6259772);