Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6368046 |
| 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 | #62618 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1499 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)(3,6)(4,12)(5,9)(7,11)(8,13)(10,15)(14,16)(17,18,22,26)(19,24,29,41)(20,25,33,43)(21,34,37,27)(23,35,39,30)(28,40,46,42)(31,48,47,38)(32,45,36,44);(1,3,6,2)(4,10,14,8)(5,9,7,11)(12,13,16,15)(17,19)(18,23)(20,31)(21,35)(22,29)(24,34)(25,42)(26,39)(27,41)(28,32)(30,37)(33,47)(36,46)(38,44)(40,43)(45,48);(1,4)(2,8)(3,10)(5,13)(6,14)(7,15)(9,16)(11,12)(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);(1,5,6,7)(2,9,3,11)(4,13,14,15)(8,16,10,12)(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);(2,3)(8,10)(9,11)(12,16)(18,26)(19,29)(25,43)(27,34)(28,46)(30,35)(31,47)(44,45)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fh,g^a=gi,d^b=df,e^b=ehi,f^b=fh,e^c=eh,g^c=gi,f^d=fh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6275571);