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 | |
4504 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10,14,16)(11,17,18,15)(12,21,23,13)(19,20,22,24)(25,26,31,28)(27,34,29,32)(30,33,38,36)(35,40,37,39);(1,2)(3,6)(4,7)(5,8)(9,11)(10,15)(12,19)(13,20)(14,18)(16,17)(21,24)(22,23)(25,27,31,29)(26,32,28,34)(30,35,38,37)(33,39,36,40);(10,16)(11,18)(13,21)(19,22)(25,26,31,28)(27,34,29,32)(30,36,38,33)(35,39,37,40);(1,3,5,4)(2,6,8,7)(9,11)(10,17)(12,22)(13,20)(14,18)(15,16)(19,23)(21,24)(25,28,31,26)(27,34,29,32)(30,36,38,33)(35,40,37,39);(1,3,5,4)(2,6,8,7)(9,12)(10,13)(11,19)(14,23)(15,20)(16,21)(17,24)(18,22)(25,29,31,27)(26,32,28,34)(30,37,38,35)(33,39,36,40);(1,4,5,3)(2,7,8,6)(9,13)(10,12)(11,20)(14,21)(15,19)(16,23)(17,22)(18,24)(25,30)(26,33)(27,35)(28,36)(29,37)(31,38)(32,39)(34,40)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=hi,d^2=gh,e^2=gh,f^2=gi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7689922);