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 | #42595 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4477 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)(12,22)(13,15)(14,18)(16,20)(19,21)(23,24)(25,26)(27,37)(28,33)(29,34)(30,35)(31,36)(32,39)(38,40);(1,3,5,8)(2,6,4,7)(9,11,16,23)(10,17,20,24)(12,21,14,13)(15,22,19,18)(25,27,31,39)(26,32,36,37)(28,35,34,38)(29,30,33,40);(1,2,5,4)(3,6,8,7)(9,12,16,14)(10,18,20,22)(11,21,23,13)(15,24,19,17)(25,28,31,34)(26,33,36,29)(27,38,39,35)(30,37,40,32);(1,4,5,2)(3,7,8,6)(9,13)(10,19)(11,12)(14,23)(15,20)(16,21)(17,18)(22,24)(25,29,31,33)(26,34,36,28)(27,30,39,40)(32,38,37,35);(9,14,16,12)(10,18,20,22)(11,13,23,21)(15,24,19,17)(25,26)(27,32)(28,33)(29,34)(30,38)(31,36)(35,40)(37,39);(9,15,16,19)(10,13,20,21)(11,22,23,18)(12,24,14,17)(25,30)(26,35)(27,33)(28,37)(29,39)(31,40)(32,34)(36,38)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=gh,d^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^e=fhi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7561443);