Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6357317 |
| 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 | |
1829 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)(3,10)(4,8)(5,14)(6,9)(7,13)(11,16)(12,15)(17,18)(19,28)(20,26)(21,27)(22,36)(23,25)(24,31)(29,43)(30,37)(32,42)(33,40)(34,39)(35,41)(38,44)(45,48)(46,47);(3,11)(7,15)(10,16)(12,13)(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);(1,3,5,13,6,11,4,12)(2,7,8,10,9,15,14,16)(18,25)(24,37)(26,39)(27,36)(28,42)(38,47)(40,41)(43,48);(1,4,6,5)(2,8,9,14)(3,12,11,13)(7,10,15,16)(17,20)(18,26)(19,29)(21,35)(22,33)(23,34)(24,38)(25,39)(27,41)(28,43)(30,46)(31,44)(32,45)(36,40)(37,47)(42,48);(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|b^2=fh,c^2=gh,d^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=ci,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6264543);