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 | |
4317 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,7)(9,10,14,17)(11,12,19,15)(13,16,22,21)(18,20,24,23)(26,31)(28,32)(29,37)(35,38);(1,2,5,4)(3,7,8,6)(9,11,14,19)(10,15,17,12)(13,18,22,24)(16,23,21,20)(26,31)(28,32)(34,39)(36,40);(1,3,5,8)(2,6,4,7)(25,26)(27,32)(28,33)(29,34)(30,31)(35,40)(36,38)(37,39);(1,4,5,2)(3,7,8,6)(9,10)(11,15)(12,19)(13,21)(14,17)(16,22)(18,20)(23,24)(25,27,30,33)(26,32,31,28)(29,38,37,35)(34,36,39,40);(9,12,14,15)(10,11,17,19)(13,20,22,23)(16,18,21,24)(25,28,30,32)(26,33,31,27)(29,36,37,40)(34,38,39,35);(9,13)(10,16)(11,18)(12,20)(14,22)(15,23)(17,21)(19,24)(25,29,30,37)(26,34,31,39)(27,35,33,38)(28,36,32,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=hi,d^2=i,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10352172);