Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574077 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
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 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2946 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 68 points:
(2,4)(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)(20,51,58,66)(24,38,54,36)(25,42,61,68)(26,44,34,43)(28,46,29,47)(30,49,35,48)(37,57,64,56)(53,67,59,63)(55,62,60,65);(5,7,10,22,12,23,11,33)(6,13,16,18,17,40,32,39)(8,24,26,19,27,54,34,21)(9,20,30,56,31,58,35,57)(14,41,43,38,45,52,44,36)(15,37,48,51,50,64,49,66)(25,60,29,53,61,55,28,59)(42,67,46,65,68,63,47,62);(1,2,3,4)(5,8)(6,14)(7,19)(9,28)(10,26)(11,34)(12,27)(13,36)(15,46)(16,43)(17,45)(18,41)(20,55)(21,23)(22,54)(24,33)(25,30)(29,31)(32,44)(35,61)(37,63)(38,40)(39,52)(42,49)(47,50)(48,68)(51,62)(53,57)(56,59)(58,60)(64,67)(65,66);(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)(22,56)(23,58)(24,60)(26,29)(27,61)(28,34)(32,49)(33,57)(36,62)(38,65)(39,66)(40,64)(41,67)(43,46)(44,47)(45,68)(52,63)(54,55);(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)(20,55)(22,24)(25,35)(28,31)(30,61)(32,44)(33,54)(36,40)(37,63)(39,41)(42,48)(46,50)(49,68)(51,62)(53,57)(56,59)(58,60)(64,67)(65,66)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=ehi,h^b=hi,d^c=dh,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, 6490397);