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 | #44312 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4323 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)(11,14)(12,15)(13,18)(16,21)(17,20)(19,23)(22,24)(25,26,31,28)(27,34,29,32)(30,36,38,33)(35,39,37,40);(1,2)(3,7)(4,5)(6,8)(9,10,14,11)(12,17,20,15)(13,16,21,18)(19,23,24,22)(25,26,31,28)(27,34,29,32)(30,33,38,36)(35,40,37,39);(1,3)(2,6)(4,7)(5,8)(10,11)(15,17)(16,18)(22,23)(25,27,31,29)(26,32,28,34)(30,35,38,37)(33,39,36,40);(9,11,14,10)(12,17,20,15)(13,18,21,16)(19,23,24,22)(25,28,31,26)(27,32,29,34)(30,36,38,33)(35,39,37,40);(1,4,5,2)(3,7,8,6)(9,12)(10,15)(11,17)(13,19)(14,20)(16,22)(18,23)(21,24)(25,29,31,27)(26,32,28,34)(30,35,38,37)(33,40,36,39);(1,3)(2,7)(4,6)(5,8)(9,13)(10,16)(11,18)(12,19)(14,21)(15,22)(17,23)(20,24)(25,30,31,38)(26,33,28,36)(27,35,29,37)(32,39,34,40)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=gh,d^2=h,e^2=gi,f^2=gh,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, 9745368);