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 | |
4344 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)(11,17)(15,18)(19,23)(22,24)(27,29)(31,33)(35,37)(39,40);(1,3,5,8)(2,6,7,4)(9,10)(11,18)(12,20)(13,14)(15,17)(16,21)(19,22)(23,24)(25,26,30,34)(27,31,29,33)(28,36,38,32)(35,40,37,39);(1,2)(3,4)(5,7)(6,8)(10,14)(15,18)(16,20)(22,24)(25,27,30,29)(26,31,34,33)(28,35,38,37)(32,39,36,40);(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)(25,28,30,38)(26,32,34,36)(27,35,29,37)(31,39,33,40);(1,4)(2,3)(5,6)(7,8)(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,29,30,27)(26,33,34,31)(28,37,38,35)(32,40,36,39)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=gh,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, 8663346);