Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437986 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2410 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4452 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,6,13,8,12,7,14)(3,15,18,33,19,34,20,9)(4,21,24,38,26,10,25,37)(5,11,28,41,29,40,30,42)(16,35,44,31,45,48,22,50)(17,39,27,55,46,57,47,32)(23,49,52,60,53,36,54,59)(43,58,61,56,62,63,51,64);(1,3)(2,9)(4,22)(5,27)(6,20)(7,18)(8,19)(10,35)(11,39)(12,33)(13,34)(14,15)(16,25)(17,28)(21,48)(23,43)(24,45)(26,44)(29,47)(30,46)(31,38)(32,41)(36,56)(37,50)(40,57)(42,55)(49,64)(51,52)(53,62)(54,61)(58,59)(60,63);(1,4,8,26)(2,10,12,21)(3,16,19,45)(5,23,29,53)(6,25,7,24)(9,31,33,50)(11,36,40,49)(13,38,14,37)(15,35,34,48)(17,43,46,62)(18,22,20,44)(27,51,47,61)(28,54,30,52)(32,56,55,64)(39,58,57,63)(41,60,42,59);(1,5)(2,11)(3,17)(4,23)(6,28)(7,30)(8,29)(9,32)(10,36)(12,40)(13,41)(14,42)(15,39)(16,43)(18,27)(19,46)(20,47)(21,49)(22,51)(24,52)(25,54)(26,53)(31,56)(33,55)(34,57)(35,58)(37,59)(38,60)(44,61)(45,62)(48,63)(50,64);(2,12)(9,33)(10,21)(11,40)(13,14)(15,34)(31,50)(32,55)(35,48)(36,49)(37,38)(39,57)(41,42)(56,64)(58,63)(59,60)Polycyclic presentation:
2^9|a^2=f,c^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,h^b=hi,f^c=fi,h^c=hi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72303711, #72303726, #72303743, … |
| maximal normal subgroup of | #72303711, #72303726, #72303743, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345183);