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 | |
4401 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)(9,10,14,16)(11,17,18,13)(12,15,20,23)(19,24,21,22)(25,26,31,37)(27,32,39,40)(28,34,38,33)(29,35,36,30);(1,3,5,8)(2,6,7,4)(9,10)(11,13)(12,15)(14,16)(17,18)(19,22)(20,23)(21,24)(25,27,31,39)(26,32,37,40)(28,36,38,29)(30,33,35,34);(1,3)(2,4)(5,8)(6,7)(11,18)(12,20)(13,17)(15,23)(25,28,31,38)(26,33,37,34)(27,29,39,36)(30,40,35,32);(1,4)(2,3)(5,6)(7,8)(9,11)(10,13)(12,21)(14,18)(15,24)(16,17)(19,20)(22,23)(25,28)(26,34)(27,36)(29,39)(30,32)(31,38)(33,37)(35,40);(9,12)(10,15)(11,19)(13,22)(14,20)(16,23)(17,24)(18,21)(25,29,31,36)(26,35,37,30)(27,38,39,28)(32,33,40,34);(9,13)(10,11)(12,22)(14,17)(15,19)(16,18)(20,24)(21,23)(25,30,31,35)(26,36,37,29)(27,33,39,34)(28,32,38,40)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=h,e^2=h,f^2=h,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, 8886545);