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 | |
4447 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,14,16)(11,15,17,12)(13,18,22,23)(19,21,24,20)(25,26,30,33)(27,28,35,31)(29,32,37,39)(34,36,40,38);(1,3,5,8)(2,6,4,7)(9,11,14,17)(10,15,16,12)(13,19,22,24)(18,21,23,20);(10,16)(11,17)(18,23)(19,24)(25,27,30,35)(26,31,33,28)(29,34,37,40)(32,38,39,36);(1,4,5,2)(3,7,8,6)(9,10)(11,15)(12,17)(13,18)(14,16)(19,21)(20,24)(22,23)(25,27)(26,28)(29,34)(30,35)(31,33)(32,36)(37,40)(38,39);(9,12,14,15)(10,17,16,11)(13,21,22,20)(18,19,23,24)(25,28)(26,27)(29,36)(30,31)(32,34)(33,35)(37,38)(39,40);(9,13)(10,18)(11,19)(12,20)(14,22)(15,21)(16,23)(17,24)(25,29,30,37)(26,32,33,39)(27,34,35,40)(28,36,31,38)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=hi,d^2=g,e^2=i,f^2=hi,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, 7854453);