Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438903 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2424 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #60387 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3026 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)(3,15)(4,21)(5,11)(6,12)(7,13)(8,14)(9,19)(10,25)(16,43)(17,39)(18,34)(20,33)(22,31)(23,49)(24,38)(26,54)(27,40)(28,42)(29,37)(30,41)(32,47)(35,46)(36,52)(44,61)(45,56)(48,57)(50,55)(51,60)(53,59)(58,63)(62,64);(1,3,6,19,8,20,7,18)(2,9,13,15,14,34,12,33)(4,22,24,16,25,45,29,46)(5,26,27,17,28,47,30,48)(10,35,38,56,21,43,37,31)(11,39,41,54,42,57,40,32)(23,44,51,62,52,63,53,50)(36,55,60,58,49,64,59,61);(1,4,6,24,8,25,7,29)(2,10,13,38,14,21,12,37)(3,16,19,45,20,46,18,22)(5,23,27,51,28,52,30,53)(9,31,15,35,34,56,33,43)(11,36,41,60,42,49,40,59)(17,44,47,62,48,63,26,50)(32,55,39,58,54,64,57,61);(1,5)(2,11)(3,17)(4,23)(6,27)(7,30)(8,28)(9,32)(10,36)(12,40)(13,41)(14,42)(15,39)(16,44)(18,26)(19,47)(20,48)(21,49)(22,50)(24,51)(25,52)(29,53)(31,55)(33,57)(34,54)(35,58)(37,59)(38,60)(43,61)(45,62)(46,63)(56,64);(1,6,8,7)(2,12,14,13)(3,18,20,19)(4,24,25,29)(5,27,28,30)(9,15,34,33)(10,37,21,38)(11,40,42,41)(16,22,46,45)(17,26,48,47)(23,51,52,53)(31,35,56,43)(32,39,54,57)(36,59,49,60)(44,50,63,62)(55,58,64,61)Polycyclic presentation:
2^9|b^2=f,c^2=f,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^b=ei,h^b=hi,g^c=gi,h^c=hi,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346101);