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 | |
4380 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)(9,10,14,15)(11,19,20,16)(12,17,23,22)(13,18,24,21)(25,26)(27,34)(28,31)(29,36)(30,33)(32,38)(35,37)(39,40);(3,7)(6,8)(10,15)(11,20)(17,22)(18,21)(26,28)(27,33)(32,36)(35,39);(1,2,5,4)(3,6,7,8)(25,27)(26,30)(28,34)(29,35)(31,33)(32,37)(36,40)(38,39);(1,3)(2,6)(4,8)(5,7)(9,11,14,20)(10,16,15,19)(12,21,23,18)(13,17,24,22)(25,28,31,26)(27,34,33,30)(29,36,38,32)(35,40,39,37);(1,4,5,2)(3,8,7,6)(9,12)(10,17)(11,21)(13,19)(14,23)(15,22)(16,24)(18,20)(25,29,31,38)(26,32,28,36)(27,35,33,39)(30,37,34,40);(9,13,14,24)(10,18,15,21)(11,22,20,17)(12,16,23,19)(25,30,31,34)(26,33,28,27)(29,37,38,40)(32,39,36,35)Polycyclic presentation:
2^9|a^2=gi,c^2=hi,d^2=g,e^2=h,f^2=g,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, 8603303);