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 | |
4738 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,16)(11,15,19,22)(12,13,20,21)(17,18,23,24)(25,26)(27,36)(28,29)(30,40)(31,33)(32,37)(34,39)(35,38);(2,3)(6,7)(9,11,14,19)(10,15,16,22)(12,17,20,23)(13,18,21,24)(25,27,31,37)(26,32,33,36)(28,38,39,30)(29,40,34,35);(1,2,5,3)(4,6,8,7)(10,16)(12,20)(15,22)(17,23)(26,33)(27,37)(28,39)(35,40);(9,12)(10,13)(11,17)(14,20)(15,18)(16,21)(19,23)(22,24)(25,28)(26,34)(27,38)(29,33)(30,37)(31,39)(32,35)(36,40);(1,3,5,2)(4,6,8,7)(9,11)(10,15)(12,17)(13,18)(14,19)(16,22)(20,23)(21,24)(25,29,31,34)(26,28,33,39)(27,35,37,40)(30,36,38,32);(1,4)(2,6)(3,7)(5,8)(9,13)(10,12)(11,18)(14,21)(15,17)(16,20)(19,24)(22,23)(25,30,31,38)(26,35,33,40)(27,28,37,39)(29,36,34,32)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=gi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9755028);