Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #466644 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16549 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
868 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,14)(3,15,20,34)(4,21,26,38)(5,27,31,11)(6,13,7,12)(9,19,35,18)(10,24,39,25)(16,44,47,32)(17,43,51,60)(22,36,46,58)(23,40,52,55)(28,37,53,61)(29,41,30,42)(33,49,54,50)(45,57,56,62)(48,64,63,59);(1,3,7,19,8,20,6,18)(2,9,13,34,14,35,12,15)(4,22,25,47,26,46,24,16)(5,17,29,49,31,51,30,50)(10,36,21,44,39,58,38,32)(11,33,41,60,27,54,42,43)(23,48,28,56,52,63,53,45)(37,59,40,62,61,64,55,57);(1,4,6,25,8,26,7,24)(2,10,12,21,14,39,13,38)(3,16,18,22,20,47,19,46)(5,28,30,52,31,53,29,23)(9,32,15,36,35,44,34,58)(11,40,42,61,27,55,41,37)(17,48,50,56,51,63,49,45)(33,59,43,62,54,64,60,57);(1,5,6,30,8,31,7,29)(2,11,12,42,14,27,13,41)(3,17,18,50,20,51,19,49)(4,23,25,28,26,52,24,53)(9,33,15,43,35,54,34,60)(10,37,21,40,39,61,38,55)(16,45,22,48,47,56,46,63)(32,57,36,59,44,62,58,64)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=f,d^2=f,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 373841);