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 | |
4644 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10,14,15)(11,19,20,16)(12,17,23,22)(13,21,24,18)(26,31)(27,33)(34,38)(35,40);(1,3,5,8)(2,6,7,4)(10,15)(12,23)(16,19)(18,21)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,40)(37,38);(1,2)(3,4)(5,7)(6,8)(10,15)(11,20)(12,23)(13,24)(27,33)(28,32)(29,37)(34,38);(1,4)(2,3)(5,6)(7,8)(9,11)(10,16)(12,18)(13,22)(14,20)(15,19)(17,24)(21,23)(25,27)(26,32)(28,31)(29,35)(30,33)(34,39)(36,38)(37,40);(1,3)(2,4)(5,8)(6,7)(9,12)(10,17)(11,21)(13,16)(14,23)(15,22)(18,20)(19,24)(25,28,30,32)(26,33,31,27)(29,36,37,39)(34,40,38,35);(1,4)(2,3)(5,6)(7,8)(9,13)(10,18)(11,22)(12,16)(14,24)(15,21)(17,20)(19,23)(25,29,30,37)(26,34,31,38)(27,35,33,40)(28,36,32,39)Polycyclic presentation:
2^9|a^2=h,b^2=i,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8886336);