Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4527 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,19)(12,14)(13,16)(15,18)(17,23)(20,24)(21,22)(25,26)(27,28)(29,37)(30,31)(32,39)(33,36)(34,38)(35,40);(1,2)(3,8)(4,6)(5,7)(9,11,15,13)(10,16,18,19)(12,20,23,22)(14,21,17,24)(25,27,30,33)(26,28,31,36)(29,34,39,40)(32,35,37,38);(1,3,5,6)(2,4,7,8)(26,31)(27,33)(32,37)(34,40);(9,12)(10,17)(11,20)(13,22)(14,18)(15,23)(16,24)(19,21)(25,26)(27,28)(29,32)(30,31)(33,36)(34,35)(37,39)(38,40);(1,2)(3,4)(5,7)(6,8)(9,13,15,11)(10,16,18,19)(12,22,23,20)(14,21,17,24)(25,28)(26,27)(29,38)(30,36)(31,33)(32,40)(34,37)(35,39);(1,4)(2,6)(3,7)(5,8)(9,14,15,17)(10,12,18,23)(11,21,13,24)(16,20,19,22)(25,29)(26,32)(27,34)(28,35)(30,39)(31,37)(33,40)(36,38)Polycyclic presentation:
2^9|b^2=gh,c^2=hi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9588865);