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 | |
4597 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)(3,4)(5,6)(7,8)(9,10,14,15)(11,12,19,16)(13,17,22,23)(18,20,21,24)(25,26)(27,33)(28,32)(29,31)(30,37)(34,38)(35,39)(36,40);(2,6)(3,7)(10,15)(12,16)(17,23)(20,24)(26,29)(32,33)(34,37)(39,40);(1,3,5,7)(2,4,6,8)(25,27,31,28)(26,32,29,33)(30,35,38,36)(34,39,37,40);(1,4)(2,7)(3,6)(5,8)(9,11,14,19)(10,16,15,12)(13,21,22,18)(17,20,23,24)(25,28,31,27)(26,33,29,32)(30,35,38,36)(34,39,37,40);(9,12,14,16)(10,11,15,19)(13,20,22,24)(17,18,23,21)(25,29,31,26)(27,33,28,32)(30,37,38,34)(35,40,36,39);(9,13)(10,17)(11,18)(12,20)(14,22)(15,23)(16,24)(19,21)(25,30,31,38)(26,34,29,37)(27,35,28,36)(32,39,33,40)Polycyclic presentation:
2^9|a^2=ghi,c^2=hi,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, 10352178);