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 | |
4360 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10,14,15)(11,16,20,23)(12,13,17,22)(18,24,21,19)(25,26,31,32)(27,33,29,28)(30,34,38,40)(35,39,37,36);(1,2,5,6)(3,7,4,8)(25,27,31,29)(26,28,32,33)(30,35,38,37)(34,36,40,39);(1,3,5,4)(2,7,6,8)(10,15)(13,22)(16,23)(19,24)(26,32)(27,29)(30,38)(36,39);(1,4,5,3)(2,8,6,7)(9,11)(10,16)(12,21)(13,19)(14,20)(15,23)(17,18)(22,24)(25,28,31,33)(26,27,32,29)(30,39,38,36)(34,37,40,35);(9,12,14,17)(10,13,15,22)(11,18,20,21)(16,24,23,19)(25,29,31,27)(26,33,32,28)(30,37,38,35)(34,39,40,36);(9,13)(10,17)(11,19)(12,15)(14,22)(16,18)(20,24)(21,23)(25,30)(26,34)(27,35)(28,36)(29,37)(31,38)(32,40)(33,39)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=gi,d^2=g,e^2=h,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7982578);