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 | |
4414 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(7,8)(9,10)(11,12)(13,15)(14,21)(16,18)(17,22)(19,24)(20,23)(25,26)(27,33)(28,35)(29,37)(30,32)(31,38)(34,36)(39,40);(1,2,6,5)(3,7,4,8)(9,11,15,16)(10,12,13,18)(14,19,22,23)(17,20,21,24)(25,26,30,32)(27,33,35,28)(29,31,38,37)(34,39,40,36);(1,3,6,4)(2,7,5,8)(9,12,15,18)(10,16,13,11)(14,20,22,24)(17,23,21,19)(25,26,30,32)(27,33,35,28)(29,31,38,37)(34,39,40,36);(1,2,6,5)(3,8,4,7)(9,12)(10,11)(13,16)(14,20)(15,18)(17,19)(21,23)(22,24)(25,27,30,35)(26,28,32,33)(29,34,38,40)(31,36,37,39);(1,4,6,3)(2,8,5,7)(9,13,15,10)(11,18,16,12)(14,21,22,17)(19,24,23,20)(25,28)(26,27)(29,36)(30,33)(31,34)(32,35)(37,40)(38,39);(1,5,6,2)(3,8,4,7)(9,14,15,22)(10,17,13,21)(11,19,16,23)(12,20,18,24)(25,29)(26,31)(27,34)(28,36)(30,38)(32,37)(33,39)(35,40)Polycyclic presentation:
2^9|b^2=gi,c^2=gi,d^2=g,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8066239);