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 | |
4583 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(9,10,14,18)(11,16,12,15)(13,17,21,24)(19,23,20,22)(25,26,31,32)(27,33,28,30)(29,37,39,34)(35,38,36,40);(3,7)(6,8)(9,11,14,12)(10,15,18,16)(13,19,21,20)(17,22,24,23)(26,32)(27,28)(34,37)(35,36);(1,2,5,4)(3,6,7,8)(25,27,31,28)(26,30,32,33)(29,35,39,36)(34,38,37,40);(1,3)(2,6)(4,8)(5,7)(9,12,14,11)(10,16,18,15)(13,19,21,20)(17,22,24,23)(25,28,31,27)(26,33,32,30)(29,36,39,35)(34,40,37,38);(1,4,5,2)(3,8,7,6)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,29,31,39)(26,34,32,37)(27,35,28,36)(30,38,33,40);(9,13,14,21)(10,17,18,24)(11,19,12,20)(15,22,16,23)(25,30,31,33)(26,27,32,28)(29,38,39,40)(34,35,37,36)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=h,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8341986);