Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6433173 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2622 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2255 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)(3,15,19,40)(4,20,25,10)(5,11,28,39)(6,13,7,12)(9,29,32,18)(16,43,21,41)(17,42,47,60)(22,34,53,48)(23,35,24,36)(26,38,27,37)(30,45,33,50)(31,54,56,46)(44,61,49,62)(51,59,52,58)(55,64,57,63);(1,3)(2,9)(4,21)(5,17)(6,18)(7,29)(8,19)(10,33)(11,31)(12,15)(13,40)(14,32)(16,25)(20,30)(22,49)(23,50)(24,45)(26,46)(27,54)(28,47)(34,57)(35,43)(36,41)(37,42)(38,60)(39,56)(44,53)(48,55)(51,64)(52,63)(58,62)(59,61);(1,4,7,24,8,25,6,23)(2,10,13,36,14,20,12,35)(3,16,18,45,19,21,29,50)(5,22,27,52,28,53,26,51)(9,30,15,41,32,33,40,43)(11,34,38,59,39,48,37,58)(17,44,46,63,47,49,54,64)(31,55,42,61,56,57,60,62);(1,5)(2,11)(3,17)(4,22)(6,26)(7,27)(8,28)(9,31)(10,34)(12,37)(13,38)(14,39)(15,42)(16,44)(18,46)(19,47)(20,48)(21,49)(23,51)(24,52)(25,53)(29,54)(30,55)(32,56)(33,57)(35,58)(36,59)(40,60)(41,61)(43,62)(45,63)(50,64);(1,6,8,7)(2,12,14,13)(3,18,19,29)(4,23,25,24)(5,26,28,27)(9,15,32,40)(10,35,20,36)(11,37,39,38)(16,45,21,50)(17,46,47,54)(22,51,53,52)(30,41,33,43)(31,42,56,60)(34,58,48,59)(44,63,49,64)(55,61,57,62)Polycyclic presentation:
2^9|a^2=i,c^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,f^a=fi,c^b=ch,f^b=fi,g^c=gi,h^c=hi,g^d=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6340372);