Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574086 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2486 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 68 points:
(1,2)(3,4)(5,6,12,17)(7,18,23,39)(8,14,27,45)(9,15,31,50)(10,32,11,16)(13,33,40,22)(19,41,21,52)(20,51,28,62)(24,38,54,36)(25,42,61,68)(26,44,34,43)(29,47,60,67)(30,49,35,48)(37,58,46,57)(53,64,59,65)(55,66,56,63);(5,7,11,33,12,23,10,22)(6,13,32,39,17,40,16,18)(8,24,34,21,27,54,26,19)(9,28,35,57,31,20,30,58)(14,41,44,36,45,52,43,38)(15,46,49,51,50,37,48,62)(25,56,60,53,61,55,29,59)(42,65,47,66,68,64,67,63);(2,4)(5,8)(6,14)(7,19)(9,29)(10,26)(11,34)(12,27)(13,36)(15,47)(16,43)(17,45)(18,41)(20,55)(21,23)(22,54)(24,33)(25,35)(28,56)(30,61)(31,60)(32,44)(37,64)(38,40)(39,52)(42,48)(46,65)(49,68)(50,67)(51,66)(53,57)(58,59)(62,63);(5,9)(6,15)(7,20)(8,25)(10,30)(11,35)(12,31)(13,37)(14,42)(16,48)(17,50)(18,51)(19,53)(21,59)(22,57)(23,28)(24,55)(26,29)(27,61)(32,49)(33,58)(34,60)(36,63)(38,66)(39,62)(40,46)(41,64)(43,67)(44,47)(45,68)(52,65)(54,56);(5,8)(6,14)(7,21)(9,29)(10,26)(11,34)(12,27)(13,38)(15,47)(16,43)(17,45)(18,52)(19,23)(20,56)(22,24)(25,35)(28,55)(30,61)(31,60)(32,44)(33,54)(36,40)(37,65)(39,41)(42,48)(46,64)(49,68)(50,67)(51,63)(53,58)(57,59)(62,66)Polycyclic presentation:
2^9|a^2=i,b^2=fi,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6481284);