Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6359034 |
| 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 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1530 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)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18,23,28)(19,29,33,44)(20,25,35,42)(21,27,22,26)(24,37,40,32)(30,45,48,39)(31,46,47,38)(34,43,36,41);(17,19,23,33)(18,24,28,40)(20,30,35,48)(21,32,22,37)(25,38,42,46)(26,29,27,44)(31,36,47,34)(39,43,45,41);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,20)(18,25)(19,30)(21,36)(22,34)(23,35)(24,38)(26,43)(27,41)(28,42)(29,45)(31,37)(32,47)(33,48)(39,44)(40,46);(17,20)(18,25)(19,31)(21,34)(22,36)(23,35)(24,39)(26,41)(27,43)(28,42)(29,46)(30,37)(32,48)(33,47)(38,44)(40,45);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,21,23,22)(18,26,28,27)(19,32,33,37)(20,34,35,36)(24,29,40,44)(25,41,42,43)(30,31,48,47)(38,39,46,45)Polycyclic presentation:
2^9|a^2=h,b^2=h,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,d^b=df,f^b=fh,e^c=eh,g^c=gi,f^d=fh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6266302);