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 | |
4361 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10)(11,16)(12,13)(14,22)(15,17)(18,23)(19,21)(20,24)(28,35)(32,36)(34,39)(37,40);(2,6)(3,7)(9,11,15,12)(10,16,17,13)(14,20,23,19)(18,21,22,24)(25,26)(27,29)(28,36)(30,31)(32,35)(33,38)(34,40)(37,39);(1,3)(2,4)(5,7)(6,8)(10,17)(11,12)(18,22)(19,20)(26,31)(27,33)(32,36)(34,39);(1,4)(2,3)(5,8)(6,7)(9,12,15,11)(10,13,17,16)(14,20,23,19)(18,21,22,24)(25,27,30,33)(26,29,31,38)(28,34,35,39)(32,37,36,40);(9,13)(10,11)(12,17)(14,21)(15,16)(18,19)(20,22)(23,24)(25,28,30,35)(26,32,31,36)(27,34,33,39)(29,37,38,40);(9,14,15,23)(10,18,17,22)(11,19,12,20)(13,21,16,24)(25,29)(26,33)(27,31)(28,37)(30,38)(32,39)(34,36)(35,40)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=gi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8664746);