Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574112 |
| 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 | |
2450 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,9,16)(3,17,6,13)(4,22,28,47)(5,12,34,45)(7,15,8,14)(10,21,37,20)(11,27,40,26)(18,50,25,35)(19,49,29,46)(23,48,54,38)(24,55,57,42)(30,39,59,64)(31,36,53,41)(32,44,33,43)(51,63,58,60)(52,62,56,61);(1,3,7,20,9,6,8,21)(2,10,14,17,16,37,15,13)(4,23,26,25,28,54,27,18)(5,29,32,53,34,19,33,31)(11,38,22,35,40,48,47,50)(12,41,43,46,45,36,44,49)(24,52,30,51,57,56,59,58)(39,61,55,63,64,62,42,60);(1,4)(2,11)(3,18)(5,30)(6,25)(7,26)(8,27)(9,28)(10,35)(12,42)(13,38)(14,22)(15,47)(16,40)(17,48)(19,52)(20,23)(21,54)(24,33)(29,56)(31,51)(32,57)(34,59)(36,61)(37,50)(39,43)(41,62)(44,64)(45,55)(46,60)(49,63)(53,58);(1,5)(2,12)(3,19)(4,24)(6,29)(7,32)(8,33)(9,34)(10,36)(11,39)(13,46)(14,43)(15,44)(16,45)(17,49)(18,51)(20,31)(21,53)(22,55)(23,56)(25,58)(26,30)(27,59)(28,57)(35,60)(37,41)(38,62)(40,64)(42,47)(48,61)(50,63)(52,54);(1,6)(2,13)(3,9)(4,25)(5,31)(7,21)(8,20)(10,14)(11,38)(12,36)(15,37)(16,17)(18,28)(19,33)(22,35)(23,27)(24,52)(26,54)(29,32)(30,51)(34,53)(39,63)(40,48)(41,45)(42,61)(43,49)(44,46)(47,50)(55,62)(56,57)(58,59)(60,64)Polycyclic presentation:
2^9|a^2=i,b^2=f,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=eh,h^b=hi,d^c=dh,g^c=gi,h^c=hi,e^d=eh,h^d=hi,h^e=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6481309);