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 | |
4726 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)(11,16)(12,18)(13,22)(14,21)(15,17)(19,23)(20,24)(28,36)(31,38)(34,40)(37,39);(1,3,5,8)(2,6,7,4)(9,11)(10,16)(12,15)(13,21)(14,22)(17,18)(19,24)(20,23)(25,26)(27,30)(28,31)(29,32)(33,35)(34,37)(36,38)(39,40);(1,3)(2,4)(5,8)(6,7)(10,17)(11,12)(19,22)(20,21)(25,26,29,32)(27,30,35,33)(28,31,36,38)(34,37,40,39);(1,4)(2,3)(5,6)(7,8)(9,12,15,11)(10,18,17,16)(13,21,23,20)(14,19,24,22)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,38)(37,40);(9,13)(10,19)(11,21)(12,20)(14,18)(15,23)(16,24)(17,22)(25,27)(26,30)(28,34)(29,35)(31,37)(32,33)(36,40)(38,39);(9,14)(10,20)(11,22)(12,19)(13,18)(15,24)(16,23)(17,21)(25,28)(26,31)(27,34)(29,36)(30,37)(32,38)(33,39)(35,40)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=ghi,d^2=h,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, 8064168);