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 | #2430 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4309 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,18)(12,22)(13,20)(14,19)(15,21)(16,23)(17,24)(25,26,31,35)(27,30,34,39)(28,36,38,32)(29,33,40,37);(1,2)(3,6)(4,7)(5,8)(9,11)(10,15)(12,20)(13,16)(14,21)(17,23)(18,19)(22,24)(25,27,31,34)(26,30,35,39)(28,37,38,33)(29,32,40,36);(1,3)(2,5)(4,8)(6,7)(9,12,14,23)(10,16,19,22)(11,20,21,17)(13,18,24,15)(25,28,31,38)(26,32,35,36)(27,33,34,37)(29,39,40,30);(3,8)(5,6)(9,12)(10,16)(11,17)(13,18)(14,23)(15,24)(19,22)(20,21)(25,29,31,40)(26,33,35,37)(27,36,34,32)(28,30,38,39);(1,2)(3,5)(4,7)(6,8)(9,13)(10,17)(11,16)(12,18)(14,24)(15,23)(19,20)(21,22)(25,26)(27,30)(28,32)(29,33)(31,35)(34,39)(36,38)(37,40);(1,3)(2,6)(4,8)(5,7)(9,11)(10,18)(12,20)(13,22)(14,21)(15,19)(16,24)(17,23)(25,30)(26,34)(27,35)(28,29)(31,39)(32,37)(33,36)(38,40)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=h,d^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8609873);