Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6357459 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#127 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1884 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,7,11)(3,4,13,15)(5,10,6,16)(8,12,14,9)(17,18,23,27)(19,28,32,42)(20,25,34,40)(21,26,22,36)(24,30,38,31)(29,41,44,48)(33,39,35,46)(37,43,47,45);(1,3,6,9,7,13,5,12)(2,8,10,4,11,14,16,15)(17,19,22,31,23,32,21,30)(18,24,26,28,27,38,36,42)(20,29,35,45,34,44,33,43)(25,37,39,41,40,47,46,48);(1,4,5,8,7,15,6,14)(2,9,16,3,11,12,10,13)(17,20,23,34)(18,25,27,40)(19,29,32,44)(21,33,22,35)(24,37,38,47)(26,39,36,46)(28,41,42,48)(30,43,31,45);(1,5,7,6)(2,10,11,16)(3,9,13,12)(4,8,15,14)(17,21,23,22)(18,26,27,36)(19,30,32,31)(20,33,34,35)(24,28,38,42)(25,39,40,46)(29,43,44,45)(37,41,47,48);(1,5,7,6)(2,10,11,16)(3,12,13,9)(4,14,15,8)(20,34)(25,40)(29,44)(33,35)(37,47)(39,46)(41,48)(43,45)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=gh,d^2=h,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=di,e^c=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6264675);