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 | #41249 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4388 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)(3,4,7,8)(9,10)(11,16)(12,18)(13,15)(14,17)(19,23)(20,24)(21,22)(26,31)(28,32)(29,37)(35,38);(2,6)(3,7)(10,13)(16,18)(17,21)(23,24)(26,31)(28,32)(34,39)(36,40);(1,3)(2,4)(5,7)(6,8)(9,11,15,12)(10,16,13,18)(14,19,22,20)(17,23,21,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,40)(36,38)(37,39);(1,4)(2,3)(5,8)(6,7)(9,12,15,11)(10,16,13,18)(14,19,22,20)(17,24,21,23)(25,27,30,33)(26,32,31,28)(29,38,37,35)(34,36,39,40);(9,13,15,10)(11,18,12,16)(14,21,22,17)(19,24,20,23)(25,28,30,32)(26,33,31,27)(29,36,37,40)(34,38,39,35);(9,14)(10,17)(11,19)(12,20)(13,21)(15,22)(16,23)(18,24)(25,29,30,37)(26,34,31,39)(27,35,33,38)(28,36,32,40)Polycyclic presentation:
2^9|a^2=hi,c^2=ghi,d^2=gi,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, 10352181);