Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32870 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2826 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14636 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
740 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)(3,9,16,34)(4,18)(5,11)(6,26)(7,13)(8,12)(10,21)(14,35,45,59)(15,31,48,58)(17,33,29,32)(19,50,46,30)(20,51)(22,38)(23,39,49,60)(24,43)(25,41)(27,37)(28,42)(36,54)(40,53)(44,61,47,64)(52,63,56,57)(55,62);(1,3,7,29,8,16,6,17)(2,9,26,33,12,34,13,32)(4,19,21,14,22,46,27,45)(5,23,24,15,25,49,28,48)(10,35,18,50,37,59,38,30)(11,39,42,58,41,60,43,31)(20,44,54,56,55,47,53,52)(36,57,51,61,40,63,62,64);(1,4,7,21,8,22,6,27)(2,10,13,38,12,37,26,18)(3,14,17,19,16,45,29,46)(5,20,24,54,25,55,28,53)(9,30,33,35,34,50,32,59)(11,40,43,51,41,36,42,62)(15,47,23,56,48,44,49,52)(31,57,39,61,58,63,60,64);(1,5)(2,11)(3,15)(4,20)(6,28)(7,24)(8,25)(9,31)(10,36)(12,41)(13,43)(14,44)(16,48)(17,23)(18,51)(19,52)(21,54)(22,55)(26,42)(27,53)(29,49)(30,57)(32,60)(33,39)(34,58)(35,61)(37,40)(38,62)(45,47)(46,56)(50,63)(59,64)Polycyclic presentation:
2^9|a^2=h,b^2=eg,c^2=g,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,c^b=cg,d^b=def,e^b=ei,g^b=gi,h^b=hi,d^c=de,f^c=fi,f^d=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 298720);