Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6367198 |
| 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 | #18480 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1509 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,7,12)(3,13,14,15)(4,9,8,16)(5,11,6,10)(17,18,23,28)(19,29,30,40)(20,25,36,43)(21,27,22,26)(24,32,38,33)(31,45,46,41)(34,48,47,39)(35,44,37,42);(1,3)(2,8)(4,12)(5,16)(6,9)(7,14)(10,15)(11,13)(17,19,23,30)(18,24,28,38)(20,34,36,47)(21,33,22,32)(25,41,43,45)(26,29,27,40)(31,37,46,35)(39,44,48,42);(1,4)(2,3)(5,15)(6,13)(7,8)(9,10)(11,16)(12,14)(19,30)(24,38)(29,40)(31,46)(32,33)(34,47)(39,48)(41,45);(1,4)(2,9)(3,10)(5,13)(6,15)(7,8)(11,14)(12,16)(17,20,23,36)(18,25,28,43)(19,31,30,46)(21,35,22,37)(24,39,38,48)(26,42,27,44)(29,45,40,41)(32,34,33,47);(17,21,23,22)(18,26,28,27)(19,32,30,33)(20,35,36,37)(24,40,38,29)(25,42,43,44)(31,34,46,47)(39,41,48,45)Polycyclic presentation:
2^9|a^2=h,b^2=hi,d^2=hi,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,f^b=fh,g^c=gi,f^d=fh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72283373, #72283489, #72283500, … |
| maximal normal subgroup of | #72283373, #72283489, #72283500, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6274714);