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 | |
4377 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)(11,12)(13,21)(14,17)(15,18)(16,23)(19,24)(20,22)(26,31)(27,34)(29,37)(36,40);(1,2)(3,4)(5,7)(6,8)(11,18)(12,15)(19,22)(20,24)(25,26)(27,32)(28,34)(29,33)(30,31)(35,40)(36,38)(37,39);(1,3)(2,6)(4,7)(5,8)(9,11,14,18)(10,15,17,12)(13,22,23,19)(16,20,21,24)(25,27,30,34)(26,32,31,28)(29,38,37,35)(33,36,39,40);(1,4,5,6)(2,3,7,8)(9,10)(11,15)(12,18)(13,16)(14,17)(19,24)(20,22)(21,23)(25,26)(27,32)(28,34)(29,33)(30,31)(35,40)(36,38)(37,39);(1,4)(2,3)(5,6)(7,8)(9,12,14,15)(10,11,17,18)(13,20,23,24)(16,19,21,22)(25,28,30,32)(26,27,31,34)(29,36,37,40)(33,35,39,38);(1,2)(3,4)(5,7)(6,8)(9,13)(10,16)(11,19)(12,20)(14,23)(15,24)(17,21)(18,22)(25,29,30,37)(26,33,31,39)(27,35,34,38)(28,36,32,40)Polycyclic presentation:
2^9|c^2=h,d^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8854907);