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 | |
4370 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)(11,13)(12,20)(14,15)(16,18)(17,22)(19,23)(21,24)(28,36)(31,38)(34,40)(37,39);(1,2,6,5)(3,7,4,8)(10,15)(11,18)(17,20)(19,24)(25,26)(27,30)(28,31)(29,32)(33,35)(34,37)(36,38)(39,40);(1,3,6,4)(2,7,5,8)(9,11)(10,16)(12,19)(13,15)(14,18)(17,23)(20,21)(22,24)(25,26,29,32)(27,30,35,33)(28,31,36,38)(34,37,40,39);(1,2,6,5)(3,8,4,7)(9,11)(10,16)(12,19)(13,15)(14,18)(17,23)(20,21)(22,24)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,38)(37,40);(1,4,6,3)(2,8,5,7)(9,12,14,22)(10,17,15,20)(11,19,18,24)(13,21,16,23)(25,27)(26,30)(28,34)(29,35)(31,37)(32,33)(36,40)(38,39);(1,5,6,2)(3,8,4,7)(9,13,14,16)(10,11,15,18)(12,21,22,23)(17,19,20,24)(25,28)(26,31)(27,34)(29,36)(30,37)(32,38)(33,39)(35,40)Polycyclic presentation:
2^9|b^2=gh,c^2=gi,d^2=gh,e^2=g,f^2=g,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, 8064159);