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 | |
4442 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)(3,7)(4,8)(5,6)(9,10)(11,22)(12,18)(13,23)(14,19)(15,20)(16,21)(17,24)(25,26,29,32)(27,30,33,36)(28,31,35,38)(34,37,39,40);(2,6)(3,8)(9,11)(10,17)(12,19)(13,15)(14,23)(16,24)(18,20)(21,22)(27,33)(30,36)(34,39)(37,40);(1,3)(2,4)(5,8)(6,7)(9,12,16,13)(10,18,21,23)(11,19,24,15)(14,17,20,22)(25,26)(27,30)(28,31)(29,32)(33,36)(34,37)(35,38)(39,40);(1,3)(2,4)(5,8)(6,7)(9,13,16,12)(10,18,21,23)(11,19,24,15)(14,22,20,17)(25,27,29,33)(26,30,32,36)(28,34,35,39)(31,37,38,40);(1,4)(2,3)(5,7)(6,8)(9,14)(10,19)(11,23)(12,17)(13,22)(15,21)(16,20)(18,24)(25,26)(27,30)(28,31)(29,32)(33,36)(34,37)(35,38)(39,40);(9,15,16,19)(10,20,21,14)(11,13,24,12)(17,18,22,23)(25,28,29,35)(26,31,32,38)(27,34,33,39)(30,37,36,40)Polycyclic presentation:
2^9|a^2=hi,c^2=gi,d^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8603994);