Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6358842 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1552 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18,22,25)(19,27,30,37)(20,24,33,39)(21,34,35,26)(23,29,38,31)(28,41,43,47)(32,45,46,40)(36,42,48,44);(17,19,22,30)(18,23,25,38)(20,28,33,43)(21,29,35,31)(24,36,39,48)(26,37,34,27)(32,42,46,44)(40,47,45,41);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,20,22,33)(18,24,25,39)(19,28,30,43)(21,32,35,46)(23,36,38,48)(26,40,34,45)(27,41,37,47)(29,42,31,44);(19,29)(21,35)(23,37)(26,34)(27,38)(28,42)(30,31)(32,46)(36,47)(40,45)(41,48)(43,44);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(18,25)(19,30)(24,39)(26,34)(28,43)(29,31)(40,45)(42,44)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=h,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,d^b=df,e^b=eh,f^b=fh,g^c=gi,f^d=fh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6266173);