Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C16 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C16 | C22×D4 |
| commutator | C4 | C23×C16 |
| Frattini, cosocle | C2×C16 | C24 |
| socle | C2 | C2×D4×C16 |
| prime radical | C2×#135 | C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16 ◁ C2×M64 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
419 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,13,7,14,28,44,5,15,25,45,26,46,56,64,8,11,29,38,30,39,58,62,27,12,40,42,57,43,32,59)(3,16,18,47,19,23,36,55,20,33,49,60,51,22,37,54,21,41,50,63,52,53,35,61,17,9,31,34,48,4,10,24);(1,3)(2,9)(4,14)(5,20)(6,18)(7,19)(8,21)(10,32)(11,33)(12,41)(13,34)(15,16)(17,27)(22,39)(23,46)(24,44)(25,49)(26,51)(28,36)(29,50)(30,52)(31,40)(35,58)(37,56)(38,60)(42,63)(43,53)(45,47)(48,57)(54,62)(55,64)(59,61);(1,4,6,24,7,16,28,47,5,23,25,55,26,33,56,60,8,22,29,54,30,41,58,63,27,53,40,61,57,9,32,34)(2,10,13,3,14,18,44,19,15,36,45,20,46,49,64,51,11,37,38,21,39,50,62,52,12,35,42,17,43,31,59,48);(2,11)(4,22)(9,33)(12,15)(13,38)(14,39)(16,41)(23,53)(24,54)(34,60)(42,45)(43,46)(44,62)(47,63)(55,61)(59,64)Polycyclic presentation:
2^9|a^2=f,c^2=f,e^2=i,f^2=g,g^2=h,h^2=i,b^a=be,d^a=di,e^a=ei,e^b=ei,d^c=di176 conjugacy classes — view the full character table.
| maximal subgroup of | #72225353, #72225380, #72225381, … |
| maximal normal subgroup of | #72225353, #72225380, #72225381, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419947);