Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6365322 |
| 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 |
| prime radical | #61659 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1561 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,9)(3,6,10,14)(4,7,12,8)(11,13,16,15)(17,18)(19,24)(20,25)(21,34)(22,26)(23,30)(27,37)(28,40)(29,41)(31,48)(32,45)(33,43)(35,39)(36,44)(38,47)(42,46);(1,3)(2,6)(4,11)(5,10)(7,13)(8,15)(9,14)(12,16)(17,19,22,29)(18,23,26,39)(20,31,33,47)(21,35,37,30)(24,27,41,34)(25,42,43,40)(28,36,46,32)(38,45,48,44);(2,7)(4,12)(6,13)(8,9)(11,16)(14,15)(17,19)(18,24)(20,28)(21,30)(22,29)(23,34)(25,40)(26,41)(27,39)(31,32)(33,46)(35,37)(36,47)(38,44)(42,43)(45,48);(17,20,22,33)(18,25,26,43)(19,28,29,46)(21,36,37,32)(23,38,39,48)(24,40,41,42)(27,45,34,44)(30,47,35,31);(3,10)(6,14)(11,16)(13,15)(18,26)(19,29)(25,43)(27,34)(28,46)(30,35)(31,47)(44,45)Polycyclic presentation:
2^9|a^2=i,b^2=h,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,f^b=fh,e^c=eh,g^c=gi,f^d=fh71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6272542);