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 | |
4507 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(25,26,31,32)(27,33,28,30)(29,37,39,34)(35,38,36,40);(2,3)(6,7)(9,10,14,13)(11,17,19,15)(12,16,21,20)(18,23,24,22)(26,32)(27,28)(34,37)(35,36);(1,3,5,2)(4,7,8,6)(10,13)(15,17)(16,20)(22,23)(25,27,31,28)(26,30,32,33)(29,35,39,36)(34,38,37,40);(1,4,5,8)(2,6,3,7)(9,11)(10,15)(12,18)(13,17)(14,19)(16,22)(20,23)(21,24)(25,28,31,27)(26,33,32,30)(29,36,39,35)(34,40,37,38);(9,12,14,21)(10,16,13,20)(11,18,19,24)(15,22,17,23)(25,29,31,39)(26,34,32,37)(27,35,28,36)(30,38,33,40);(9,13,14,10)(11,17,19,15)(12,20,21,16)(18,23,24,22)(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=g,d^2=g,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8104461);