Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33657 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#481 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | C2×#1926 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
765 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,8,13,7,30)(3,9,17,34,15,32,18,50)(4,19,22,38,20,52,23,56)(5,24,27,39,29,58,28,11)(10,26,36,21,35,55,40,53)(14,41,44,62,42,51,45,64)(16,46,48,37,25,57,49,33)(31,47,61,43,60,59,63,54);(1,3,6,17,8,15,7,18)(2,9,12,34,13,32,30,50)(4,14,23,45,20,42,22,44)(5,25,27,49,29,16,28,48)(10,31,40,63,35,60,36,61)(11,37,24,57,39,33,58,46)(19,51,56,62,52,41,38,64)(21,54,26,59,53,43,55,47);(1,4,7,22,8,20,6,23)(2,10,12,40,13,35,30,36)(3,14,18,44,15,42,17,45)(5,26,28,21,29,55,27,53)(9,31,34,63,32,60,50,61)(11,38,24,19,39,56,58,52)(16,47,49,43,25,59,48,54)(33,62,46,41,37,64,57,51);(3,15)(9,32)(14,42)(16,25)(17,18)(31,60)(33,37)(34,50)(41,51)(43,54)(44,45)(46,57)(47,59)(48,49)(61,63)(62,64)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=eh,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=di,f^b=fi,e^c=ei,f^c=fh,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72285705, #72285725, #72285937, … |
| maximal normal subgroup of | #72285705, #72285725, #72285937, … |
| maximal quotient of | #72310157, #72310172, #72321903, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402079);