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 | #62868 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2634 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,10,35)(3,17,36,61)(4,23,37,29)(5,12,38,26)(6,14,34,44)(7,13,31,47)(8,15,9,16)(11,39,60,20)(18,48,62,58)(19,49,63,51)(21,45,22,50)(24,28,25,27)(30,43,46,33)(32,41,57,42)(40,64,53,54)(52,56,59,55);(1,3)(2,11)(4,18)(5,19)(6,20)(7,32)(8,22)(9,21)(10,36)(12,40)(13,17)(14,45)(15,42)(16,41)(23,56)(24,51)(25,49)(26,53)(27,48)(28,58)(29,55)(30,54)(31,57)(33,59)(34,39)(35,60)(37,62)(38,63)(43,52)(44,50)(46,64)(47,61);(1,4)(2,5)(3,18)(6,24)(7,33)(8,26)(9,12)(10,37)(11,19)(13,27)(14,46)(15,23)(16,29)(17,48)(20,51)(21,40)(22,53)(25,34)(28,47)(30,44)(31,43)(32,59)(35,38)(36,62)(39,49)(41,55)(42,56)(45,64)(50,54)(52,57)(58,61)(60,63);(1,5)(2,12)(3,19)(4,16)(6,30)(7,28)(8,23)(9,29)(10,38)(11,40)(13,25)(14,43)(15,37)(17,49)(18,41)(20,54)(21,55)(22,56)(24,47)(26,35)(27,31)(32,58)(33,44)(34,46)(36,63)(39,64)(42,62)(45,52)(48,57)(50,59)(51,61)(53,60);(1,6,9,7)(2,13,16,14)(3,20,21,32)(4,24,12,33)(5,27,29,46)(8,31,10,34)(11,17,41,45)(15,44,35,47)(18,51,40,59)(19,48,55,64)(22,57,36,39)(23,30,38,28)(25,26,43,37)(42,50,60,61)(49,53,52,62)(54,63,58,56)Polycyclic presentation:
2^9|a^2=hi,e^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, 6259757);