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:
(1,2,5,6)(3,4,7,8)(9,10)(11,19)(12,21)(13,18)(14,15)(16,20)(17,22)(23,24)(25,26)(27,32)(28,39)(29,33)(30,37)(31,35)(34,36)(38,40);(2,6)(3,7)(10,15)(12,22)(16,19)(18,23)(25,27,31,38)(26,32,35,40)(28,36,33,30)(29,37,39,34);(1,3)(2,4)(5,7)(6,8)(25,28,31,33)(26,29,35,39)(27,36,38,30)(32,37,40,34);(1,4)(2,3)(5,8)(6,7)(9,11)(10,16)(12,23)(13,21)(14,20)(15,19)(17,24)(18,22)(25,26)(27,32)(28,29)(30,34)(31,35)(33,39)(36,37)(38,40);(9,12)(10,17)(11,18)(13,16)(14,22)(15,21)(19,24)(20,23)(25,29)(26,33)(27,37)(28,35)(30,32)(31,39)(34,38)(36,40);(9,13)(10,18)(11,17)(12,16)(14,24)(15,23)(19,22)(20,21)(25,30)(26,34)(27,28)(29,32)(31,36)(33,38)(35,37)(39,40)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10319416);