Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6352341 |
| 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 | |
2587 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(2,9)(3,16)(6,31)(10,40)(11,36)(12,37)(13,38)(14,32)(15,39)(17,21)(18,23)(20,53)(28,57)(34,55)(35,43)(41,50)(42,44)(45,62)(46,52)(47,63)(48,59)(49,64)(51,61)(56,60);(1,2)(3,10)(4,11)(5,12)(6,32)(7,14)(8,15)(9,31)(13,27)(16,44)(17,41)(18,36)(19,40)(20,45)(21,46)(22,35)(23,59)(24,48)(25,49)(26,42)(28,61)(29,51)(30,52)(33,62)(34,38)(37,57)(39,55)(43,53)(47,58)(50,54)(56,64)(60,63);(1,3,7,16)(2,10,14,44)(4,21,24,17)(5,18,29,23)(6,33,31,22)(8,20,27,53)(9,35,32,62)(11,46,48,41)(12,36,51,59)(13,43,15,45)(19,55,26,34)(25,57,58,28)(30,56,54,60)(37,47,61,49)(38,40,39,42)(50,63,52,64);(1,4,8,25)(2,11,15,49)(3,17,20,28)(5,26,30,22)(6,23,34,60)(7,24,27,58)(9,36,39,64)(10,41,45,61)(12,42,52,35)(13,47,14,48)(16,21,53,57)(18,55,56,31)(19,54,33,29)(32,59,38,63)(37,44,46,43)(40,50,62,51);(1,5,27,54)(2,12,13,50)(3,18,53,60)(4,22,58,19)(6,28,55,21)(7,29,8,30)(9,37,38,41)(10,36,43,63)(11,35,47,40)(14,51,15,52)(16,23,20,56)(17,31,57,34)(24,33,25,26)(32,61,39,46)(42,48,62,49)(44,59,45,64)Polycyclic presentation:
2^9|c^2=h,d^2=i,e^2=hi,f^2=h,b^a=bf,c^a=cg,f^a=fh,f^b=fh,g^b=gi,d^c=dh,f^c=fi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6259568);