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 | |
4520 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,16)(11,18,13,15)(12,17,22,21)(19,24,20,23)(25,26)(27,36)(28,30)(29,34)(31,35)(32,39)(33,38)(37,40);(1,2)(3,8)(4,6)(5,7)(9,11,14,13)(10,15,16,18)(12,20,22,19)(17,24,21,23)(25,27)(26,32)(28,37)(29,38)(30,34)(31,39)(33,40)(35,36);(1,3)(2,4)(5,6)(7,8)(10,16)(15,18)(17,21)(23,24)(25,28,31,38)(26,30,35,33)(27,37,39,29)(32,34,36,40);(3,6)(4,8)(9,10)(11,15)(12,21)(13,18)(14,16)(17,22)(19,24)(20,23)(25,28)(26,33)(27,37)(29,39)(30,35)(31,38)(32,40)(34,36);(1,4,5,8)(2,6,7,3)(9,12,14,22)(10,17,16,21)(11,19,13,20)(15,23,18,24)(25,29,31,37)(26,34,35,40)(27,38,39,28)(30,36,33,32);(1,2)(3,4)(5,7)(6,8)(9,13,14,11)(10,18,16,15)(12,19,22,20)(17,23,21,24)(25,30,31,33)(26,28,35,38)(27,34,39,40)(29,32,37,36)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=gh,e^2=gi,f^2=g,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, 9068148);