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 | |
4421 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)(3,7,8,4)(9,10)(11,13)(12,20)(14,16)(15,18)(17,23)(19,24)(21,22)(28,36)(32,37)(34,40)(38,39);(3,8)(4,7)(9,11,14,18)(10,15,16,13)(12,21,23,19)(17,22,20,24)(26,30)(31,33)(32,37)(38,39);(1,3)(2,4)(5,8)(6,7)(10,16)(11,18)(17,20)(19,21)(25,26,29,30)(27,33,35,31)(28,32,36,37)(34,39,40,38);(1,4)(2,3)(5,7)(6,8)(9,12,14,23)(10,17,16,20)(11,19,18,21)(13,22,15,24);(9,10)(11,15)(12,17)(13,18)(14,16)(19,24)(20,23)(21,22)(25,27,29,35)(26,31,30,33)(28,34,36,40)(32,38,37,39);(9,13,14,15)(10,18,16,11)(12,22,23,24)(17,21,20,19)(25,28)(26,32)(27,34)(29,36)(30,37)(31,38)(33,39)(35,40)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=hi,d^2=g,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8002315);