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 | |
4394 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(11,17)(15,18)(19,23)(22,24)(25,26,31,29)(27,32,30,38)(28,35,37,33)(34,40,36,39);(1,3,5,8)(2,6,7,4)(9,10)(11,18)(12,20)(13,14)(15,17)(16,21)(19,22)(23,24)(25,27,31,30)(26,32,29,38)(28,34,37,36)(33,39,35,40);(1,2)(3,4)(5,7)(6,8)(10,14)(15,18)(16,20)(22,24)(25,28)(26,33)(27,34)(29,35)(30,36)(31,37)(32,39)(38,40);(1,4)(2,3)(5,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24);(1,3)(2,4)(5,8)(6,7)(9,11,13,17)(10,15,14,18)(12,19,21,23)(16,22,20,24)(25,29,31,26)(27,32,30,38)(28,35,37,33)(34,39,36,40);(1,4)(2,3)(5,6)(7,8)(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,30,31,27)(26,32,29,38)(28,36,37,34)(33,39,35,40)Polycyclic presentation:
2^9|a^2=h,b^2=h,e^2=hi,f^2=gh,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, 8663295);