Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #361285 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2290 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13897 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
683 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)(3,15,19,39)(4,10,23,36)(5,24,26,37)(6,12)(7,14)(8,13)(9,18,32,28)(11,25,38,27)(16,41)(17,46)(20,30)(21,50)(22,35,29,40)(31,48)(33,45)(34,54)(42,53)(43,62,64,55)(44,56)(47,59)(49,61,63,57)(51,58)(52,60);(1,3,8,28,7,19,6,18)(2,9,12,15,14,32,13,39)(4,20,29,45,23,44,22,16)(5,17,27,53,26,48,25,47)(10,33,35,30,36,41,40,56)(11,31,24,42,38,46,37,59)(21,49,54,64,52,63,51,43)(34,57,50,55,58,61,60,62);(1,4,6,22,7,23,8,29)(2,10,12,35,14,36,13,40)(3,16,18,44,19,45,28,20)(5,21,25,51,26,52,27,54)(9,30,15,41,32,56,39,33)(11,34,24,50,38,58,37,60)(17,43,47,63,48,64,53,49)(31,55,42,61,46,62,59,57);(1,5,7,26)(2,11)(3,17,19,48)(4,21,23,52)(6,25,8,27)(9,31)(10,34)(12,24)(13,37)(14,38)(15,42)(16,43,45,64)(18,47,28,53)(20,49,44,63)(22,51,29,54)(30,55)(32,46)(33,57)(35,50)(36,58)(39,59)(40,60)(41,61)(56,62)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=e,d^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,g^b=gi,h^b=hi,g^c=gi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 268602);