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 |
| prime radical | #62867 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2700 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,8,14)(3,15,19,45)(4,10,24,38)(5,11,27,12)(6,28,26,41)(7,13,31,40)(9,17,34,18)(16,44,47,43)(20,42,50,33)(21,36,25,39)(22,52,30,55)(23,37,54,62)(29,35,51,61)(32,49,58,46)(48,57,64,60)(53,59,56,63);(1,3)(2,9)(4,16)(5,17)(6,29)(7,20)(8,19)(10,32)(11,33)(12,42)(13,35)(14,34)(15,28)(18,27)(21,46)(22,53)(23,48)(24,47)(25,49)(26,51)(30,56)(31,50)(36,57)(37,59)(38,58)(39,60)(40,61)(41,45)(43,55)(44,52)(54,64)(62,63);(1,4,7,23)(2,10,13,37)(3,16,20,48)(5,21,26,22)(6,30,27,25)(8,24,31,54)(9,32,35,59)(11,36,41,52)(12,39,28,55)(14,38,40,62)(15,43,42,60)(17,46,51,53)(18,49,29,56)(19,47,50,64)(33,57,45,44)(34,58,61,63);(1,4)(2,10)(3,16)(5,25)(6,22)(7,23)(8,24)(9,32)(11,39)(12,36)(13,37)(14,38)(15,44)(17,49)(18,46)(19,47)(20,48)(21,27)(26,30)(28,52)(29,53)(31,54)(33,60)(34,58)(35,59)(40,62)(41,55)(42,57)(43,45)(50,64)(51,56)(61,63);(1,5,7,26)(2,11,13,41)(3,17,20,51)(4,21,23,22)(6,8,27,31)(9,33,35,45)(10,36,37,52)(12,40,28,14)(15,34,42,61)(16,46,48,53)(18,50,29,19)(24,25,54,30)(32,57,59,44)(38,39,62,55)(43,58,60,63)(47,49,64,56)Polycyclic presentation:
2^9|a^2=i,c^2=h,e^2=h,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, 6259610);