Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6353061 |
| 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 |
| prime radical | #62963 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2676 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,8,15)(3,17,23,50)(4,5,27,11)(6,13,32,14)(7,34,33,46)(9,16,37,24)(10,20,42,21)(12,44,28,30)(18,48,54,58)(19,55,38,52)(22,47,59,40)(25,29,45,43)(26,61,36,31)(35,41,60,51)(39,63,53,56)(49,64,62,57);(1,3)(2,10)(4,18)(5,19)(6,20)(7,35)(8,23)(9,22)(11,38)(12,39)(13,40)(14,47)(15,42)(16,41)(17,34)(21,32)(24,51)(25,52)(26,62)(27,54)(28,53)(29,56)(30,57)(31,58)(33,60)(36,49)(37,59)(43,63)(44,64)(45,55)(46,50)(48,61);(1,4,9,28)(2,11,16,44)(3,18,22,53)(5,24,30,15)(6,25,33,26)(7,36,32,45)(8,27,37,12)(10,38,41,64)(13,43,46,31)(14,29,34,61)(17,48,47,56)(19,51,57,42)(20,52,60,62)(21,55,35,49)(23,54,59,39)(40,63,50,58);(1,5)(2,12)(3,19)(4,24)(6,31)(7,29)(8,11)(9,30)(10,39)(13,25)(14,45)(15,28)(16,27)(17,49)(18,51)(20,58)(21,48)(22,57)(23,38)(26,46)(32,61)(33,43)(34,36)(35,56)(37,44)(40,52)(41,54)(42,53)(47,55)(50,62)(59,64)(60,63);(1,6,9,33)(2,13,16,46)(3,20,22,60)(4,25,28,26)(5,29,30,61)(7,8,32,37)(10,40,41,50)(11,43,44,31)(12,36,27,45)(14,24,34,15)(17,42,47,51)(18,52,53,62)(19,56,57,48)(21,59,35,23)(38,63,64,58)(39,49,54,55)Polycyclic presentation:
2^9|a^2=i,c^2=h,e^2=h,f^2=h,b^a=bf,c^a=cg,d^a=dh,f^a=fh,f^b=fh,g^b=gi,f^c=fi,e^d=ehi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6260399);