Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #361285 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2290 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13901 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
949 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,13,7,14,6,12)(3,15,18,33,19,32,20,9)(4,10,23,36,24,38,25,37)(5,26,27,39,28,40,29,11)(16,34,44,30,43,56,21,49)(17,45,48,41,47,31,46,57)(22,51,54,35,53,59,52,60)(42,61,50,64,62,55,63,58);(1,3,8,20,7,19,6,18)(2,9,12,15,14,33,13,32)(4,21,25,16,24,44,23,43)(5,17,29,48,28,47,27,46)(10,34,36,49,38,56,37,30)(11,31,26,41,39,45,40,57)(22,50,54,42,53,63,52,62)(35,58,51,61,60,64,59,55);(1,4,6,23,7,24,8,25)(2,10,12,36,14,38,13,37)(3,16,18,21,19,43,20,44)(5,22,27,52,28,53,29,54)(9,30,15,34,33,49,32,56)(11,35,26,51,39,60,40,59)(17,42,46,50,47,62,48,63)(31,55,41,58,45,61,57,64);(1,5,8,29,7,28,6,27)(2,11,12,26,14,39,13,40)(3,17,20,48,19,47,18,46)(4,22,25,54,24,53,23,52)(9,31,15,41,33,45,32,57)(10,35,36,51,38,60,37,59)(16,42,44,63,43,62,21,50)(30,55,34,58,49,61,56,64)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=e,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,g^b=gi,h^b=hi,g^c=gi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 268758);