Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #140203 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1934 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C2 | #11189 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
643 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,7,13,8,15,33,14)(3,16,21,46,22,36,34,37)(4,23,27,43,29,58,28,61)(5,12,30,11,31,47,6,32)(9,20,38,18,40,56,39,19)(10,41,44,25,45,26,48,60)(17,50,53,35,55,57,54,63)(24,42,52,62,59,64,51,49);(1,3)(2,9)(4,24)(5,20)(6,18)(7,34)(8,22)(10,42)(11,46)(12,36)(13,39)(14,38)(15,40)(16,47)(17,26)(19,30)(21,33)(23,57)(25,53)(27,51)(28,52)(29,59)(31,56)(32,37)(35,43)(41,55)(44,49)(45,64)(48,62)(50,58)(54,60)(61,63);(1,4,5,25,7,28,30,41,8,29,31,60,33,27,6,26)(2,10,11,23,14,44,12,43,15,45,32,58,13,48,47,61)(3,17,18,51,21,54,56,59,22,55,19,52,34,53,20,24)(9,35,16,49,39,57,37,42,40,63,36,62,38,50,46,64)Polycyclic presentation:
2^9|a^2=g,c^2=d,d^2=h,e^2=h,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,h^a=hi,c^b=cf,d^b=dhi,e^b=eh,f^b=fh,g^b=gi,h^b=hi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72307185, #72307216, #72307253, … |
| maximal normal subgroup of | #72307185, #72307216, #72307253, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55729);