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 | #36235 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4381 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,7,8,4)(9,10,15,19)(11,20,21,16)(12,17,22,24)(13,14,18,23)(25,26,30,34)(27,35,36,31)(28,37,39,32)(29,33,38,40);(1,3)(2,4)(5,8)(6,7)(11,21)(12,22)(16,20)(17,24)(25,27,30,36)(26,31,34,35)(28,38,39,29)(32,40,37,33);(9,11)(10,16)(12,18)(13,22)(14,17)(15,21)(19,20)(23,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,36)(37,39)(38,40);(2,6)(4,7)(9,12,15,22)(10,17,19,24)(11,18,21,13)(14,20,23,16)(25,28,30,39)(26,32,34,37)(27,29,36,38)(31,33,35,40);(1,3)(2,4)(5,8)(6,7)(9,13,15,18)(10,14,19,23)(11,22,21,12)(16,17,20,24);(1,4)(2,3)(5,7)(6,8)(9,14)(10,18)(11,17)(12,16)(13,19)(15,23)(20,22)(21,24)(25,29)(26,33)(27,28)(30,38)(31,32)(34,40)(35,37)(36,39)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,d^2=h,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8353030);