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 | |
4703 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(26,31)(27,35)(33,37)(34,36);(3,7)(6,8)(9,10,14,15)(11,16,18,23)(12,13,17,22)(19,24,21,20)(25,26,30,31)(27,32,35,40)(28,33,38,37)(29,34,39,36);(1,2,5,4)(3,6,7,8)(10,15)(11,18)(13,22)(19,21)(27,35)(29,39)(32,40)(34,36);(9,11)(10,16)(12,21)(13,20)(14,18)(15,23)(17,19)(22,24)(25,27)(26,32)(28,34)(29,37)(30,35)(31,40)(33,39)(36,38);(1,3,5,7)(2,6,4,8)(9,12,14,17)(10,13,15,22)(11,19,18,21)(16,24,23,20)(25,28)(26,33)(27,36)(29,32)(30,38)(31,37)(34,35)(39,40);(1,4,5,2)(3,8,7,6)(9,13)(10,17)(11,20)(12,15)(14,22)(16,19)(18,24)(21,23)(25,29)(26,34)(27,37)(28,32)(30,39)(31,36)(33,35)(38,40)Polycyclic presentation:
2^9|b^2=gi,c^2=h,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8826429);