Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6792444 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #38904 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1783 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(5,21)(8,27)(10,33)(13,34)(14,37)(15,40)(18,42)(22,48)(24,49)(30,55)(39,47)(41,51)(43,60)(45,61)(54,64)(57,62);(1,2)(3,13)(4,9)(5,22)(6,25)(7,11)(8,17)(10,24)(12,23)(14,35)(15,29)(16,34)(18,43)(19,32)(20,46)(21,48)(26,27)(28,39)(30,45)(31,44)(33,49)(36,51)(37,56)(38,54)(40,53)(41,50)(42,60)(47,52)(55,61)(57,63)(58,64)(59,62);(1,3)(2,8)(4,14)(5,15)(6,26)(7,16)(9,28)(10,29)(11,27)(12,34)(13,25)(17,23)(18,38)(19,37)(20,47)(21,40)(22,50)(24,51)(30,54)(31,56)(32,52)(33,53)(35,46)(36,48)(39,44)(41,49)(42,58)(43,62)(45,63)(55,64)(57,60)(59,61);(1,4,7,19)(2,9,11,32)(3,14,16,37)(5,18,21,42)(6,20,23,44)(8,28,27,52)(10,30,33,55)(12,31,25,46)(13,35,34,56)(15,38,40,58)(17,39,26,47)(22,43,48,60)(24,45,49,61)(29,54,53,64)(36,57,50,62)(41,59,51,63);(1,5,23,49,7,21,6,24)(2,10,25,48,11,33,12,22)(3,15,17,41,16,40,26,51)(4,18,44,61,19,42,20,45)(8,29,13,36,27,53,34,50)(9,30,46,60,32,55,31,43)(14,38,39,59,37,58,47,63)(28,54,35,57,52,64,56,62)Polycyclic presentation:
2^9|d^2=i,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,g^b=gi,h^b=hi,f^c=fi,h^c=hi,g^d=gi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6699643);