Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #396371 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14732 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1375 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,13,8,14,7,12)(3,15,18,9,19,33,30,34)(4,20,24,38,25,37,23,10)(5,26,27,39,29,11,28,40)(16,42,21,50,45,31,44,35)(17,46,47,56,48,41,58,32)(22,52,54,36,55,49,53,57)(43,59,51,60,63,61,62,64);(1,3,6,18,8,19,7,30)(2,9,12,15,14,34,13,33)(4,21,23,16,25,44,24,45)(5,17,27,47,29,48,28,58)(10,35,20,42,38,50,37,31)(11,32,39,41,26,56,40,46)(22,51,53,43,55,62,54,63)(36,60,49,61,57,64,52,59);(1,4,8,25)(2,10,14,38)(3,16,19,45)(5,22,29,55)(6,23,7,24)(9,31,34,42)(11,36,26,57)(12,20,13,37)(15,35,33,50)(17,43,48,63)(18,44,30,21)(27,53,28,54)(32,59,56,61)(39,49,40,52)(41,60,46,64)(47,62,58,51);(1,5,6,27,8,29,7,28)(2,11,12,39,14,26,13,40)(3,17,18,47,19,48,30,58)(4,22,23,53,25,55,24,54)(9,32,15,41,34,56,33,46)(10,36,20,49,38,57,37,52)(16,43,44,62,45,63,21,51)(31,59,35,60,42,61,50,64)Polycyclic presentation:
2^9|a^2=f,b^2=e,c^2=i,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^b=gi,f^c=fi,g^c=gi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 303568);