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 | #18334 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2561 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,10,5)(3,17,23,52)(4,24,29,45)(6,33,26,46)(7,36,39,48)(8,15,9,16)(11,41,19,21)(12,28,30,27)(13,34,31,35)(14,38,32,25)(18,53,57,63)(20,42,60,58)(22,47,40,62)(37,44,64,49)(43,56,50,61)(51,55,54,59);(1,3,9,22)(2,11,16,44)(4,18,28,56)(5,19,15,49)(6,20,35,59)(7,37,25,21)(8,40,10,23)(12,42,45,51)(13,43,33,63)(14,47,36,17)(24,54,30,58)(26,60,34,55)(27,61,29,57)(31,50,46,53)(32,62,48,52)(38,41,39,64);(1,4,9,28)(2,12,16,45)(3,18,22,56)(5,30,15,24)(6,25,35,7)(8,27,10,29)(11,42,44,51)(13,32,33,48)(14,46,36,31)(17,50,47,53)(19,58,49,54)(20,21,59,37)(23,57,40,61)(26,38,34,39)(41,55,64,60)(43,62,63,52);(1,5,10,2)(3,19,23,11)(4,24,29,45)(6,13,26,31)(7,32,39,14)(8,16,9,15)(12,28,30,27)(17,21,52,41)(18,54,57,51)(20,43,60,50)(22,49,40,44)(25,48,38,36)(33,34,46,35)(37,62,64,47)(42,56,58,61)(53,59,63,55);(1,6,10,26)(2,13,5,31)(3,20,23,60)(4,25,29,38)(7,27,39,28)(8,34,9,35)(11,43,19,50)(12,32,30,14)(15,46,16,33)(17,51,52,54)(18,21,57,41)(22,59,40,55)(24,36,45,48)(37,61,64,56)(42,62,58,47)(44,63,49,53)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=h,d^2=i,e^2=i,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, 6259744);