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 | |
4497 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)(11,17)(12,22)(13,18)(14,16)(15,19)(20,23)(21,24)(25,26,32,36)(27,29,30,33)(28,37,38,34)(31,35,40,39);(1,2)(3,5)(4,6)(7,8)(9,11,16,19)(10,17,14,15)(12,21,13,20)(18,23,22,24)(25,27)(26,33)(28,35)(29,36)(30,32)(31,37)(34,40)(38,39);(2,3)(6,7)(9,12,16,13)(10,18,14,22)(11,20,19,21)(15,23,17,24)(25,28,32,38)(26,34,36,37)(27,35,30,39)(29,31,33,40);(4,8)(6,7)(9,13,16,12)(10,18,14,22)(11,21,19,20)(15,23,17,24)(25,29)(26,27)(28,31)(30,36)(32,33)(34,35)(37,39)(38,40);(1,3,5,2)(4,7,8,6)(9,14,16,10)(11,17,19,15)(12,18,13,22)(20,23,21,24)(25,30,32,27)(26,29,36,33)(28,35,38,39)(31,34,40,37);(1,4)(2,6)(3,7)(5,8)(9,15,16,17)(10,19,14,11)(12,23,13,24)(18,21,22,20)(25,31)(26,35)(27,37)(28,33)(29,38)(30,34)(32,40)(36,39)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=h,d^2=gh,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8521318);