Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32870 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2826 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14669 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
706 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,14,8,12,7,13)(3,9,18,34,17,35,30,33)(4,19,23,10,22,38,24,39)(5,11,28,42,29,41,27,43)(15,44,20,31,46,36,52,49)(16,32,25,40,47,59,48,60)(21,50,54,37,26,55,53,62)(45,63,51,57,56,61,64,58);(1,3,7,18,8,17,6,30)(2,9,14,33,12,35,13,34)(4,20,24,46,22,52,23,15)(5,25,28,16,29,48,27,47)(10,36,38,31,39,44,19,49)(11,40,43,59,41,60,42,32)(21,45,54,64,26,56,53,51)(37,57,50,61,62,58,55,63);(1,4,6,23,8,22,7,24)(2,10,13,19,12,39,14,38)(3,15,18,20,17,46,30,52)(5,26,27,54,29,21,28,53)(9,31,33,44,35,49,34,36)(11,37,42,55,41,62,43,50)(16,45,48,64,47,56,25,51)(32,58,40,63,59,57,60,61);(1,5)(2,11)(3,16)(4,21)(6,28)(7,27)(8,29)(9,32)(10,37)(12,41)(13,43)(14,42)(15,45)(17,47)(18,25)(19,50)(20,51)(22,26)(23,54)(24,53)(30,48)(31,57)(33,60)(34,40)(35,59)(36,61)(38,55)(39,62)(44,63)(46,56)(49,58)(52,64)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=f,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=def,e^b=ei,f^b=fi,g^b=gi,d^c=de,e^c=ei,f^d=fi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299189);