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)(3,11)(4,20)(5,10)(6,13)(7,14)(8,17)(9,24)(12,31)(15,32)(16,44)(18,34)(19,45)(21,35)(22,40)(23,39)(25,38)(26,57)(27,41)(28,46)(29,42)(30,43)(33,51)(36,55)(37,48)(47,59)(49,53)(50,60)(52,61)(54,62)(56,63)(58,64);(1,3,6,17,7,19,31,18)(2,8,12,11,14,34,13,45)(4,21,23,49,25,48,24,15)(5,16,28,51,30,52,29,27)(9,35,38,32,39,37,20,53)(10,33,42,44,43,41,46,61)(22,54,26,58,56,50,55,47)(36,62,63,59,57,60,40,64);(1,4)(2,9)(3,15)(5,26)(6,23)(7,25)(8,32)(10,40)(11,37)(12,38)(13,20)(14,39)(16,50)(17,21)(18,48)(19,49)(22,29)(24,31)(27,58)(28,56)(30,55)(33,60)(34,53)(35,45)(36,42)(41,62)(43,63)(44,64)(46,57)(47,51)(52,54)(59,61);(1,5)(2,10)(3,16)(4,22)(6,28)(7,30)(8,33)(9,36)(11,44)(12,42)(13,46)(14,43)(15,47)(17,51)(18,27)(19,52)(20,40)(21,54)(23,26)(24,55)(25,56)(29,31)(32,59)(34,41)(35,62)(37,60)(38,63)(39,57)(45,61)(48,50)(49,58)(53,64);(1,3)(2,11)(4,15)(5,27)(6,17)(7,19)(8,13)(9,37)(10,41)(12,34)(14,45)(16,28)(18,31)(20,32)(21,23)(22,54)(24,48)(25,49)(26,58)(29,52)(30,51)(33,43)(35,39)(36,59)(38,53)(40,62)(42,61)(44,46)(47,55)(50,56)(57,64)(60,63)Polycyclic presentation:
2^9|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,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, 6481303);