Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29902 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2622 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14265 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
943 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,28)(3,14,19,38,17,9,18,31)(4,20,23,33,24,50,22,10)(5,11,27,34,26,35,25,54)(15,39,43,29,41,48,42,36)(16,37,47,59,45,30,46,57)(21,49,53,58,52,64,51,32)(40,60,62,55,44,63,61,56);(1,3,6,18,8,17,7,19)(2,9,12,31,13,14,28,38)(4,15,22,42,24,41,23,43)(5,16,25,46,26,45,27,47)(10,29,20,48,33,36,50,39)(11,30,34,57,35,37,54,59)(21,40,51,61,52,44,53,62)(32,55,49,63,58,56,64,60);(1,4,6,22,8,24,7,23)(2,10,12,20,13,33,28,50)(3,15,18,42,17,41,19,43)(5,21,25,51,26,52,27,53)(9,29,31,48,14,36,38,39)(11,32,34,49,35,58,54,64)(16,44,46,62,45,40,47,61)(30,56,57,60,37,55,59,63);(1,5)(2,11)(3,16)(4,21)(6,25)(7,27)(8,26)(9,30)(10,32)(12,34)(13,35)(14,37)(15,40)(17,45)(18,46)(19,47)(20,49)(22,51)(23,53)(24,52)(28,54)(29,55)(31,57)(33,58)(36,56)(38,59)(39,60)(41,44)(42,61)(43,62)(48,63)(50,64)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=f,f^2=i,h^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,e^b=ei,d^c=de,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 287565);