Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6347544 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2681 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(2,8)(3,13)(4,17)(5,21)(9,28)(10,32)(11,25)(12,22)(14,35)(15,40)(16,43)(18,27)(20,46)(23,49)(26,31)(29,52)(30,56)(33,58)(34,50)(36,38)(37,39)(41,54)(42,61)(44,55)(47,51)(53,60)(57,59)(62,63);(1,2)(3,9)(4,18)(5,22)(6,12)(7,11)(8,21)(10,17)(13,31)(14,29)(15,41)(16,44)(19,27)(20,47)(23,50)(24,34)(25,49)(26,39)(28,43)(30,38)(32,45)(33,46)(35,55)(36,60)(37,52)(40,57)(42,53)(48,51)(54,61)(56,63)(58,64)(59,62);(3,14)(5,23)(8,25)(9,29)(13,35)(15,38)(16,43)(19,48)(21,49)(22,50)(26,52)(27,51)(28,44)(30,41)(31,55)(32,58)(36,40)(37,39)(42,63)(45,64)(53,56)(54,59)(57,60)(61,62);(1,3)(2,9)(4,15)(5,16)(6,14)(7,13)(8,26)(10,30)(11,31)(12,29)(17,38)(18,41)(19,42)(20,36)(21,39)(22,44)(23,37)(24,35)(25,28)(27,53)(32,59)(33,57)(34,55)(40,46)(43,49)(45,62)(47,60)(48,61)(50,52)(51,54)(56,58)(63,64);(1,4,7,20)(2,10,11,33)(3,15,13,36)(5,19,23,48)(6,17,24,46)(8,27,25,51)(9,30,31,57)(12,18,34,47)(14,38,35,40)(16,42,37,61)(21,45,49,64)(22,32,50,58)(26,53,28,54)(29,41,55,60)(39,62,43,63)(44,59,52,56)Polycyclic presentation:
2^9|e^2=i,f^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,e^b=eh,f^b=fh,g^b=gi,d^c=dh,f^c=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6254769);