Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6792404 |
| 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:
(8,27)(13,35)(14,38)(29,53)(37,49)(39,59)(40,48)(60,63);(1,2,7,11)(3,13,16,35)(4,9,19,32)(5,21,24,50)(6,25,22,12)(8,26,27,17)(10,33,34,23)(14,36,38,57)(15,29,41,53)(18,43,45,62)(20,47,44,31)(28,48,52,40)(30,55,56,46)(37,42,49,51)(39,54,59,64)(58,60,61,63);(1,3)(2,8)(4,14)(5,15)(6,26)(7,16)(9,28)(10,29)(11,27)(12,35)(13,25)(17,22)(18,39)(19,38)(20,48)(21,49)(23,51)(24,41)(30,54)(31,57)(32,52)(33,42)(34,53)(36,47)(37,50)(40,44)(43,61)(45,59)(46,63)(55,60)(56,64)(58,62);(1,4)(2,9)(3,14)(5,18)(6,20)(7,19)(8,28)(10,30)(11,32)(12,31)(13,36)(15,39)(16,38)(17,40)(21,43)(22,44)(23,46)(24,45)(25,47)(26,48)(27,52)(29,54)(33,55)(34,56)(35,57)(37,58)(41,59)(42,60)(49,61)(50,62)(51,63)(53,64);(1,5,22,33,7,24,6,23)(2,10,25,50,11,34,12,21)(3,15,17,42,16,41,26,51)(4,18,44,55,19,45,20,46)(8,29,13,37,27,53,35,49)(9,30,47,62,32,56,31,43)(14,39,40,60,38,59,48,63)(28,54,36,58,52,64,57,61)Polycyclic presentation:
2^9|b^2=i,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,g^b=gi,h^b=hi,f^c=fi,h^c=hi,g^d=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6699602);