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 | |
4436 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)(11,17)(12,16)(13,18)(14,15)(19,24)(20,21)(22,23)(25,26)(27,30)(28,33)(29,39)(31,35)(32,36)(34,37)(38,40);(2,5)(6,7)(9,11)(10,13)(12,20)(14,18)(15,17)(16,24)(19,22)(21,23)(25,27)(26,32)(28,34)(29,33)(30,35)(31,36)(37,40)(38,39);(1,2)(3,6)(4,5)(7,8)(10,15)(11,18)(16,23)(19,20)(25,28,31,38)(26,33,35,40)(27,34,36,39)(29,30,37,32);(9,11)(10,13)(12,19)(14,18)(15,17)(16,21)(20,22)(23,24)(25,29,31,37)(26,34,35,39)(27,33,36,40)(28,30,38,32);(1,2)(3,7)(4,5)(6,8)(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,24)(18,20)(25,30)(26,27)(28,37)(29,38)(31,32)(33,34)(35,36)(39,40);(1,3)(2,6)(4,8)(5,7)(9,13)(10,11)(12,21)(14,17)(15,18)(16,19)(20,23)(22,24)(25,26)(27,32)(28,33)(29,34)(30,36)(31,35)(37,39)(38,40)Polycyclic presentation:
2^9|c^2=gh,d^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, 9686454);