Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6792416 |
| 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 | |
1785 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)(3,9)(4,19)(5,25)(6,30)(7,12)(8,13)(10,23)(11,29)(14,35)(15,45)(16,40)(17,33)(18,34)(20,49)(21,51)(22,41)(24,37)(26,55)(27,39)(28,57)(31,44)(32,47)(36,54)(38,46)(42,62)(43,60)(48,52)(50,59)(53,56)(58,64)(61,63);(1,3,8,18)(2,9,13,34)(4,20,22,44)(5,26,29,46)(6,17,7,16)(10,35,37,60)(11,38,25,55)(12,40,30,33)(14,23,43,24)(15,27,47,28)(19,31,41,49)(21,42,52,63)(32,57,45,39)(36,59,56,64)(48,62,51,61)(50,54,58,53);(1,4)(2,10)(3,14)(5,21)(6,24)(7,23)(8,22)(9,31)(11,36)(12,41)(13,37)(15,42)(16,20)(17,44)(18,43)(19,30)(25,56)(26,58)(27,54)(28,53)(29,52)(32,59)(33,60)(34,49)(35,40)(38,62)(39,51)(45,64)(46,50)(47,63)(48,57)(55,61);(1,2)(3,9)(4,10)(5,11)(6,30)(7,12)(8,13)(14,31)(15,32)(16,40)(17,33)(18,34)(19,24)(20,35)(21,36)(22,37)(23,41)(25,29)(26,38)(27,57)(28,39)(42,59)(43,49)(44,60)(45,47)(46,55)(48,54)(50,61)(51,53)(52,56)(58,62)(63,64);(1,5,6,27,8,29,7,28)(2,11,30,57,13,25,12,39)(3,15,16,46,18,47,17,26)(4,21,24,54,22,52,23,53)(9,32,40,55,34,45,33,38)(10,36,19,48,37,56,41,51)(14,42,20,50,43,63,44,58)(31,59,35,61,49,64,60,62)Polycyclic presentation:
2^9|b^2=i,e^2=fhi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,e^b=efh,g^b=gi,h^b=hi,f^c=fi,g^c=gi,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, 6699615);