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 | |
4419 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,19)(11,20,18,14)(12,23,22,16)(13,17,21,24)(25,26)(27,35)(28,33)(29,34)(30,31)(32,36)(37,40)(38,39);(1,2)(3,8)(4,6)(5,7)(9,11,15,18)(10,14,19,20)(12,21,22,13)(16,24,23,17)(27,36)(29,37)(32,35)(34,40);(1,3)(2,4)(5,6)(7,8)(9,12,15,22)(10,16,19,23)(11,21,18,13)(14,24,20,17)(25,27,31,36)(26,32,30,35)(28,37,39,29)(33,34,38,40);(3,6)(4,8)(9,12)(10,16)(11,21)(13,18)(14,24)(15,22)(17,20)(19,23)(25,28,31,39)(26,33,30,38)(27,29,36,37)(32,40,35,34);(1,4,5,8)(2,6,7,3)(9,13,15,21)(10,17,19,24)(11,22,18,12)(14,23,20,16)(25,29)(26,34)(27,28)(30,40)(31,37)(32,38)(33,35)(36,39);(1,2)(3,4)(5,7)(6,8)(9,14)(10,18)(11,19)(12,24)(13,16)(15,20)(17,22)(21,23)(25,30,31,26)(27,35,36,32)(28,38,39,33)(29,34,37,40)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=h,d^2=gh,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9068912);