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 | |
4560 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,15)(12,20)(13,14)(16,22)(17,19)(18,24)(21,23)(25,26,30,32)(27,35,36,31)(28,33,38,40)(29,39,37,34);(1,2)(3,8)(4,6)(5,7)(9,10)(11,17)(12,16)(13,14)(15,19)(18,24)(20,22)(21,23)(25,27,30,36)(26,31,32,35)(28,37,38,29)(33,39,40,34);(1,3,5,6)(2,4,7,8)(10,14)(15,17)(16,20)(23,24)(26,32)(31,35)(33,40)(34,39);(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,22)(21,24)(25,26)(27,31)(28,33)(29,34)(30,32)(35,36)(37,39)(38,40);(1,2)(3,4)(5,7)(6,8)(9,11,13,19)(10,15,14,17)(12,21,22,18)(16,24,20,23)(25,28,30,38)(26,33,32,40)(27,29,36,37)(31,34,35,39);(1,4)(2,6)(3,7)(5,8)(9,12,13,22)(10,16,14,20)(11,18,19,21)(15,23,17,24)(25,29)(26,34)(27,28)(30,37)(31,33)(32,39)(35,40)(36,38)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=ghi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9591254);