Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6357957 |
| 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 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1815 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)(3,13)(4,9)(5,10)(6,11)(7,12)(8,14)(15,16)(17,18,23,25)(19,28,32,42)(20,26,34,39)(21,27,22,36)(24,30,37,31)(29,43,45,48)(33,40,35,41)(38,46,47,44);(1,2,6,10,7,12,5,11)(3,8,14,13,15,4,9,16)(17,19,21,30,23,32,22,31)(18,24,36,42,25,37,27,28)(20,29,35,46,34,45,33,44)(26,38,40,48,39,47,41,43);(1,3,5,9,7,15,6,14)(2,8,11,16,12,4,10,13)(18,25)(24,37)(26,39)(27,36)(28,42)(38,47)(40,41)(43,48);(1,4,7,8)(2,9,12,14)(3,11,15,10)(5,13,6,16)(17,20,23,34)(18,26,25,39)(19,29,32,45)(21,35,22,33)(24,38,37,47)(27,41,36,40)(28,43,42,48)(30,46,31,44);(1,5,7,6)(2,10,12,11)(3,9,15,14)(4,16,8,13)(17,21,23,22)(18,27,25,36)(19,30,32,31)(20,33,34,35)(24,28,37,42)(26,40,39,41)(29,44,45,46)(38,48,47,43)Polycyclic presentation:
2^9|a^2=hi,b^2=fh,c^2=gh,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,e^b=ei,d^c=di,e^d=eh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6265216);