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 |
| prime radical | #2688 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4182 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)(11,18)(12,15)(13,20)(14,17)(16,22)(19,23)(21,24)(25,26)(27,35)(28,38)(29,33)(30,34)(31,37)(32,39)(36,40);(1,3,5,8)(2,6,4,7)(9,11)(10,14)(12,18)(13,19)(15,17)(16,23)(20,21)(22,24)(25,27,30,37)(26,31,34,35)(28,36,39,29)(32,40,38,33);(1,2,5,4)(3,6,8,7)(10,12)(11,17)(16,20)(19,24)(25,26,30,34)(27,31,37,35)(28,32,39,38)(29,33,36,40);(9,12,15,10)(11,18,17,14)(13,20,22,16)(19,21,24,23)(25,26)(27,31)(28,32)(29,33)(30,34)(35,37)(36,40)(38,39);(1,2,5,4)(3,6,8,7)(9,13)(10,16)(11,19)(12,20)(14,23)(15,22)(17,24)(18,21)(25,28)(26,32)(27,36)(29,37)(30,39)(31,40)(33,35)(34,38);(1,4,5,2)(3,7,8,6)(9,14,15,18)(10,17,12,11)(13,21,22,23)(16,19,20,24)(25,29,30,36)(26,33,34,40)(27,28,37,39)(31,32,35,38)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=gi,d^2=i,e^2=gh,f^2=g,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, 7854248);