Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7157001 |
| 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 |
| prime radical | #61108 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1705 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,14,9,16,8,15)(3,17,21,44,23,54,22,10)(4,24,28,11,30,46,29,47)(5,12,34,50,36,51,35,55)(6,37,39,53,41,52,40,13)(18,42,25,62,49,33,61,45)(19,48,31,27,57,63,56,43)(20,58,59,64,60,26,38,32);(1,3)(2,10)(4,25)(5,31)(6,38)(7,22)(8,21)(9,23)(11,45)(12,48)(13,32)(14,54)(15,17)(16,44)(18,28)(19,34)(20,40)(24,42)(26,37)(27,55)(29,49)(30,61)(33,46)(35,57)(36,56)(39,60)(41,59)(43,50)(47,62)(51,63)(52,58)(53,64);(1,4)(2,11)(3,18,23,49)(5,32,36,64)(6,27)(7,29)(8,28)(9,30)(10,42,44,33)(12,38,51,59)(13,19)(14,24)(15,46)(16,47)(17,45,54,62)(20,50,60,55)(21,61,22,25)(26,35,58,34)(31,52)(37,56)(39,48)(40,63)(41,43)(53,57);(1,5,7,34,9,36,8,35)(2,12,14,50,16,51,15,55)(3,19,21,31,23,57,22,56)(4,26,28,32,30,58,29,64)(6,33,39,45,41,42,40,62)(10,43,17,48,44,27,54,63)(11,20,46,59,47,60,24,38)(13,49,37,61,53,18,52,25);(1,6,7,39,9,41,8,40)(2,13,15,52,16,53,14,37)(3,20,21,59,23,60,22,38)(4,27,29,48,30,43,28,63)(5,33,34,45,36,42,35,62)(10,26,54,64,44,58,17,32)(11,19,46,31,47,57,24,56)(12,49,55,25,51,18,50,61)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=eh,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=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7064199);