Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6360760 |
| 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 | |
1516 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,10)(4,8)(5,13)(6,9)(7,12)(11,16)(14,15)(17,18,23,29)(19,25,32,41)(20,26,35,44)(21,28,22,27)(24,36,39,31)(30,40,47,42)(33,48,46,38)(34,45,37,43);(3,11)(7,15)(10,16)(12,14)(17,19)(18,24)(20,33)(21,31)(22,36)(23,32)(25,27)(26,42)(28,41)(29,39)(30,34)(35,46)(37,47)(38,43)(40,44)(45,48);(1,3)(2,7)(4,12)(5,14)(6,11)(8,16)(9,15)(10,13)(17,19)(18,25)(20,30)(21,36)(22,31)(23,32)(24,27)(26,40)(28,39)(29,41)(33,34)(35,47)(37,46)(38,43)(42,44)(45,48);(17,20,23,35)(18,26,29,44)(19,30,32,47)(21,34,22,37)(24,38,39,48)(25,40,41,42)(27,43,28,45)(31,46,36,33);(1,4,6,5)(2,8,9,13)(3,12,11,14)(7,16,15,10)(17,21,23,22)(18,27,29,28)(19,31,32,36)(20,34,35,37)(24,25,39,41)(26,43,44,45)(30,46,47,33)(38,40,48,42)Polycyclic presentation:
2^9|a^2=h,d^2=h,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,c^b=ci,d^b=df,f^b=fh,e^c=eh,g^c=gi,f^d=fh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6268069);