Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7157007 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #44936 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1712 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,13,9,15,8,14)(3,16,20,40,21,39,35,10)(4,6,24,11,26,34,25,33)(5,12,30,44,31,46,36,45)(17,37,48,19,50,32,22,47)(18,42,27,57,54,64,53,38)(23,29,55,41,56,43,28,60)(49,63,61,52,58,59,51,62);(1,3)(2,10)(4,22)(5,27)(6,32)(7,35)(8,20)(9,21)(11,19)(12,42)(13,39)(14,16)(15,40)(17,25)(18,30)(23,49)(24,50)(26,48)(28,61)(29,62)(31,53)(33,47)(34,37)(36,54)(38,44)(41,59)(43,52)(45,57)(46,64)(51,55)(56,58)(60,63);(1,4)(2,11)(3,17,21,50)(5,28,31,55)(6,13)(7,25)(8,24)(9,26)(10,37,40,32)(12,43,46,29)(14,34)(15,33)(16,47,39,19)(18,51)(20,22,35,48)(23,30,56,36)(27,58)(38,52)(41,45,60,44)(42,63)(49,53)(54,61)(57,62)(59,64);(1,5,7,30,9,31,8,36)(2,12,13,44,15,46,14,45)(3,18,20,27,21,54,35,53)(4,23,24,55,26,56,25,28)(6,29,11,41,34,43,33,60)(10,38,16,42,40,57,39,64)(17,49,48,61,50,58,22,51)(19,52,32,59,47,62,37,63);(1,6,8,33,9,34,7,11)(2,4,13,24,15,26,14,25)(3,19,35,37,21,47,20,32)(5,29,36,60,31,43,30,41)(10,17,16,48,40,50,39,22)(12,23,44,55,46,56,45,28)(18,52,53,63,54,62,27,59)(38,49,42,61,57,58,64,51)Polycyclic presentation:
2^9|a^2=f,c^2=fg,d^2=f,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ehi,h^a=hi,c^b=ch,d^b=df,e^b=eh,f^b=fi,h^b=hi,d^c=dh,f^c=fi,g^c=gi,g^d=gi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7064205);