Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6356901 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1499 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,7,9)(3,12,13,16)(4,10,8,15)(5,11,6,14)(17,18,23,28)(19,29,31,41)(20,34,37,43)(21,26,22,27)(24,33,40,32)(25,38,44,36)(30,42,35,39)(45,47,48,46);(1,3,5,15,7,13,6,10)(2,8,14,16,9,4,11,12)(17,19,21,32,23,31,22,33)(18,24,27,29,28,40,26,41)(20,35,36,47,37,30,38,46)(25,42,34,48,44,39,43,45);(2,9)(3,13)(10,15)(11,14)(17,20,23,37)(18,25,28,44)(19,30,31,35)(21,36,22,38)(24,39,40,42)(26,43,27,34)(29,45,41,48)(32,46,33,47);(1,4)(2,10)(3,14)(5,12)(6,16)(7,8)(9,15)(11,13)(19,31)(24,40)(29,41)(30,35)(32,33)(39,42)(45,48)(46,47);(1,5,7,6)(2,11,9,14)(3,15,13,10)(4,16,8,12)(17,21,23,22)(18,26,28,27)(19,32,31,33)(20,36,37,38)(24,41,40,29)(25,43,44,34)(30,46,35,47)(39,48,42,45)Polycyclic presentation:
2^9|a^2=h,b^2=fhi,c^2=i,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,c^b=ch,d^b=dh,g^c=gi,e^d=ehi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6264257);