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 | |
4541 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)(11,17)(12,19)(13,15)(14,16)(18,21)(20,23)(22,24)(26,28)(31,33)(32,36)(38,39);(1,3,5,8)(2,6,4,7)(9,11)(10,12)(13,18)(14,19)(15,20)(16,17)(21,24)(22,23);(9,10,14,16)(11,17,19,12)(13,15,22,24)(18,21,23,20)(25,26,30,28)(27,33,35,31)(29,32,37,36)(34,39,40,38);(1,2,5,4)(3,6,8,7)(9,10)(11,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27,30,35)(26,31,28,33)(29,34,37,40)(32,38,36,39);(1,2,5,4)(3,6,8,7)(9,12)(10,11)(13,21)(14,17)(15,23)(16,19)(18,24)(20,22)(25,28,30,26)(27,33,35,31)(29,36,37,32)(34,39,40,38);(1,4,5,2)(3,7,8,6)(9,13)(10,15)(11,18)(12,20)(14,22)(16,24)(17,21)(19,23)(25,29,30,37)(26,32,28,36)(27,34,35,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=gi,b^2=gi,c^2=h,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7766496);