Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574107 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | D4×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2460 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)(3,16)(4,21)(5,11)(6,12)(7,14)(8,15)(9,19)(10,26)(13,33)(17,35)(18,48)(20,37)(22,38)(23,42)(24,53)(25,41)(27,40)(28,58)(29,36)(30,46)(31,43)(32,44)(34,45)(39,55)(47,50)(49,59)(51,60)(52,61)(54,62)(56,64)(57,63);(1,3,7,19,8,6,33,20)(2,9,13,16,15,37,14,12)(4,22,25,24,27,50,26,17)(5,18,30,29,32,34,31,52)(10,38,40,35,41,47,21,53)(11,36,43,48,44,61,46,45)(23,54,28,57,56,51,55,49)(39,62,64,59,58,60,42,63);(1,4)(2,10)(3,17)(5,28)(6,24)(7,25)(8,27)(9,35)(11,42)(12,38)(13,40)(14,21)(15,41)(16,47)(18,51)(19,22)(20,50)(23,31)(26,33)(29,49)(30,56)(32,55)(34,54)(36,60)(37,53)(39,43)(44,64)(45,59)(46,58)(48,63)(52,57)(61,62);(1,5,8,32)(2,11,15,44)(3,18,6,34)(4,23,27,56)(7,30,33,31)(9,36,37,61)(10,39,41,58)(12,45,16,48)(13,43,14,46)(17,49,24,57)(19,29,20,52)(21,42,40,64)(22,54,50,51)(25,28,26,55)(35,59,53,63)(38,62,47,60);(1,6)(2,12)(3,8)(4,24)(5,29)(7,20)(9,13)(10,38)(11,36)(14,37)(15,16)(17,27)(18,31)(19,33)(21,53)(22,26)(23,51)(25,50)(28,49)(30,34)(32,52)(35,40)(39,63)(41,47)(42,60)(43,48)(44,61)(45,46)(54,56)(55,57)(58,59)(62,64)Polycyclic presentation:
2^9|b^2=f,d^2=i,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,h^b=hi,d^c=dh,g^c=gi,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, 6481306);