Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30871 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2675 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14402 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
912 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,14,6,13)(3,15,19,33,20,41,18,9)(4,21,25,36,26,42,24,10)(5,11,28,39,30,40,29,38)(16,45,47,34,48,31,22,43)(17,44,27,56,50,37,49,32)(23,51,53,62,55,61,54,35)(46,60,57,59,64,58,52,63);(1,3,6,18,8,20,7,19)(2,9,12,15,14,33,13,41)(4,22,24,48,26,47,25,16)(5,27,28,50,30,49,29,17)(10,34,21,31,36,43,42,45)(11,37,38,56,40,44,39,32)(23,46,53,57,55,64,54,52)(35,60,61,63,62,58,51,59);(1,4,6,24,8,26,7,25)(2,10,12,21,14,36,13,42)(3,16,18,22,20,48,19,47)(5,23,29,54,30,55,28,53)(9,31,15,43,33,45,41,34)(11,35,39,51,40,62,38,61)(17,46,49,52,50,64,27,57)(32,58,44,63,56,60,37,59);(1,5,6,29,8,30,7,28)(2,11,12,39,14,40,13,38)(3,17,18,49,20,50,19,27)(4,23,24,54,26,55,25,53)(9,32,15,44,33,56,41,37)(10,35,21,51,36,62,42,61)(16,46,22,52,48,64,47,57)(31,58,43,63,45,60,34,59)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=f,d^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,g^b=gi,g^c=gi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 292401);