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 | |
4384 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)(11,19)(12,18)(13,16)(14,20)(15,17)(21,22)(23,24)(25,26)(27,36)(28,38)(29,40)(30,35)(31,32)(33,39)(34,37);(1,3,5,8)(2,6,4,7)(9,11,15,13)(10,16,17,19)(12,21,20,23)(14,22,18,24)(25,27,31,35)(26,30,32,36)(28,37,39,29)(33,34,38,40);(10,17)(12,20)(16,19)(21,23)(26,32)(27,35)(28,39)(34,40);(1,2,5,4)(3,6,8,7)(9,12,15,20)(10,18,17,14)(11,21,13,23)(16,24,19,22)(25,28)(26,33)(27,37)(29,35)(30,34)(31,39)(32,38)(36,40);(9,13,15,11)(10,19,17,16)(12,23,20,21)(14,24,18,22)(25,29,31,37)(26,34,32,40)(27,28,35,39)(30,38,36,33);(1,4,5,2)(3,7,8,6)(9,14,15,18)(10,20,17,12)(11,22,13,24)(16,23,19,21)(25,30)(26,35)(27,32)(28,34)(29,33)(31,36)(37,38)(39,40)Polycyclic presentation:
2^9|a^2=gh,b^2=g,d^2=gi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7803214);