Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33359 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×#99 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C2×#41 | C24 |
| socle | C2 | C2×C4×#99 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×M16 ◁ C2×#568 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1183 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,11,7,12,24,34)(3,13,16,36,17,37,42,58)(4,9,20,31,18,29,44,53)(5,21,22,47,23,10,32,33)(8,25,27,41,28,49,52,15)(14,35,40,57,38,56,60,64)(19,45,46,61,43,30,54,55)(26,48,51,59,50,62,63,39);(1,3)(2,8)(4,14)(5,15,23,41)(6,16)(7,17)(9,26)(10,13,21,37)(11,27)(12,28)(18,38)(19,39,43,59)(20,40)(22,25,32,49)(24,42)(29,50)(30,35,45,56)(31,51)(33,36,47,58)(34,52)(44,60)(46,48,54,62)(53,63)(55,57,61,64);(1,4,6,20,7,18,24,44)(2,9,11,31,12,29,34,53)(3,14,16,40,17,38,42,60)(5,19,22,46,23,43,32,54)(8,26,27,51,28,50,52,63)(10,30,33,55,21,45,47,61)(13,35,36,57,37,56,58,64)(15,39,25,48,41,59,49,62);(4,18)(9,29)(14,38)(19,43)(20,44)(26,50)(30,45)(31,53)(35,56)(39,59)(40,60)(46,54)(48,62)(51,63)(55,61)(57,64)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,e^2=i,f^2=i,b^a=be,e^a=ei,g^a=ghi,e^b=eh,h^b=hi,d^c=di,g^e=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72312695, #72312756, #72312796, … |
| maximal normal subgroup of | #72312695, #72312756, #72312796, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400224);