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 | #3498 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1581 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)(17,18,24,30)(19,31,36,42)(20,26,21,27)(22,29,23,28)(25,34,43,35)(32,47,37,41)(33,44,48,40)(38,46,39,45);(1,3)(2,6)(4,11)(5,10)(7,13)(8,15)(9,14)(12,16)(17,19,24,36)(18,25,30,43)(20,37,21,32)(22,35,23,34)(26,44,27,40)(28,31,29,42)(33,38,48,39)(41,45,47,46);(2,7)(4,12)(6,13)(8,9)(11,16)(14,15)(17,20,24,21)(18,26,30,27)(19,32,36,37)(22,39,23,38)(25,40,43,44)(28,46,29,45)(31,47,42,41)(33,35,48,34);(17,21,24,20)(18,27,30,26)(19,33,36,48)(22,39,23,38)(25,41,43,47)(28,46,29,45)(31,44,42,40)(32,35,37,34);(3,10)(6,14)(11,16)(13,15)(17,22,24,23)(18,28,30,29)(19,34,36,35)(20,38,21,39)(25,42,43,31)(26,45,27,46)(32,48,37,33)(40,47,44,41)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=h,d^2=h,e^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, 6272559);