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 |
| prime radical | #62864 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2669 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,12)(3,13,16,38)(4,9,19,33)(5,21,23,50)(6,10,25,34)(8,26,29,15)(11,35,37,24)(14,39,41,61)(17,36,43,60)(18,30,44,31)(20,47,46,57)(22,28,51,56)(27,52,55,42)(32,49,59,45)(40,58,63,62)(48,53,64,54);(1,3,6,17)(2,8,10,28)(4,14,18,40)(5,22,24,15)(7,16,25,43)(9,27,30,53)(11,36,21,13)(12,29,34,56)(19,41,44,63)(20,48,45,42)(23,51,35,26)(31,54,33,55)(32,58,47,39)(37,60,50,38)(46,64,49,52)(57,61,59,62);(4,18)(5,23)(9,30)(11,37)(13,38)(14,40)(15,26)(19,44)(20,49)(21,50)(22,51)(24,35)(27,53)(31,33)(32,57)(36,60)(39,62)(41,63)(42,64)(45,46)(47,59)(48,52)(54,55)(58,61);(1,4)(2,9)(3,14)(5,20)(6,18)(7,19)(8,27)(10,30)(11,32)(12,33)(13,39)(15,42)(16,41)(17,40)(21,47)(22,48)(23,46)(24,45)(25,44)(26,52)(28,53)(29,55)(31,34)(35,49)(36,58)(37,59)(38,61)(43,63)(50,57)(51,64)(54,56)(60,62);(2,10)(4,19)(8,28)(9,31)(11,21)(12,34)(13,36)(14,41)(18,44)(20,46)(27,54)(29,56)(30,33)(32,57)(37,50)(38,60)(39,62)(40,63)(42,52)(45,49)(47,59)(48,64)(53,55)(58,61)Polycyclic presentation:
2^9|a^2=i,b^2=h,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, 6259722);