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 | |
4381 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,15,19)(11,20,21,16)(12,13,17,23)(14,18,24,22)(25,26)(27,32)(28,30)(29,35)(31,36)(33,38)(34,37)(39,40);(1,3,5,8)(2,6,4,7)(9,11)(10,16)(12,14)(13,22)(15,21)(17,24)(18,23)(19,20)(25,27,31,38)(26,32,36,33)(28,29,37,39)(30,35,34,40);(11,21)(14,24)(16,20)(18,22)(25,27,31,38)(26,33,36,32)(28,29,37,39)(30,40,34,35);(9,12)(10,13)(11,14)(15,17)(16,22)(18,20)(19,23)(21,24)(25,28)(26,34)(27,29)(30,36)(31,37)(32,40)(33,35)(38,39);(1,4,5,2)(3,7,8,6)(9,13)(10,17)(11,22)(12,19)(14,20)(15,23)(16,24)(18,21)(25,29)(26,35)(27,37)(28,38)(30,33)(31,39)(32,34)(36,40);(9,14)(10,18)(11,12)(13,20)(15,24)(16,23)(17,21)(19,22)(25,30,31,34)(26,28,36,37)(27,35,38,40)(29,33,39,32)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=h,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,e^d=ehi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7558950);