Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13074 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1893 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C2 | #10721 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
863 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,6,12,8,14,26,13)(3,15,19,43,18,42,50,44)(4,10,22,36,23,39,53,38)(5,24,27,11,7,29,28,40)(9,30,33,20,32,51,34,17)(16,45,48,64,47,41,62,63)(21,52,54,35,25,55,56,37)(31,58,60,49,59,57,61,46);(1,3,7,20,26,50,27,30,8,18,5,17,6,19,28,51)(2,9,11,15,12,33,29,43,14,32,40,42,13,34,24,44)(4,16,21,46,53,62,56,57,23,47,25,49,22,48,54,58)(10,31,37,41,36,60,52,63,39,59,35,45,38,61,55,64);(1,4)(2,10)(3,16)(5,25)(6,22)(7,21)(8,23)(9,31)(11,37)(12,36)(13,38)(14,39)(15,41)(17,49)(18,47)(19,48)(20,46)(24,55)(26,53)(27,56)(28,54)(29,52)(30,57)(32,59)(33,60)(34,61)(35,40)(42,45)(43,63)(44,64)(50,62)(51,58)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=hi,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,h^a=hi,e^b=ei,d^c=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72307198, #72307245, #72307295, … |
| maximal normal subgroup of | #72307198, #72307245, #72307295, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53644);