Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6362118 |
| 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 | #61901 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1601 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)(4,7)(5,8)(9,12)(10,13)(11,14)(15,16)(17,18,23,27)(19,28,31,38)(20,25,22,35)(21,26,34,43)(24,36,40,29)(30,45,47,42)(32,41,37,44)(33,48,46,39);(1,3,5,11)(2,6,8,14)(4,10,12,16)(7,13,9,15)(17,19,23,31)(18,24,27,40)(20,29,22,36)(21,33,34,46)(25,38,35,28)(26,42,43,45)(30,37,47,32)(39,41,48,44);(2,7)(4,12)(6,13)(8,9)(10,16)(14,15)(17,20,23,22)(18,25,27,35)(19,29,31,36)(21,32,34,37)(24,38,40,28)(26,41,43,44)(30,46,47,33)(39,42,48,45);(17,21,23,34)(18,26,27,43)(19,30,31,47)(20,32,22,37)(24,39,40,48)(25,41,35,44)(28,45,38,42)(29,46,36,33);(2,8)(6,14)(7,9)(13,15)(18,27)(24,40)(25,35)(26,43)(28,38)(39,48)(41,44)(42,45)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=h,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fh,g^a=gi,d^b=df,f^b=fh,g^c=gi,f^d=fh65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6269341);