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 | |
4413 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(4,8)(9,10,15,19)(11,16,20,23)(12,17,21,22)(13,14,18,24)(25,26)(27,35)(28,30)(29,33)(31,32)(34,38)(36,37)(39,40);(1,2,6,5)(3,8,7,4)(9,11,15,20)(10,16,19,23)(12,13,21,18)(14,22,24,17)(26,32)(28,38)(33,35)(36,39);(1,3)(2,4)(5,8)(6,7)(11,20)(12,21)(16,23)(17,22)(25,27,31,29)(26,33,32,35)(28,39,38,36)(30,37,34,40);(2,5)(3,7)(9,12,15,21)(10,17,19,22)(11,13,20,18)(14,23,24,16)(25,28,31,38)(26,34,32,30)(27,36,29,39)(33,37,35,40);(1,4)(2,3)(5,7)(6,8)(9,13)(10,14)(11,21)(12,20)(15,18)(16,22)(17,23)(19,24)(25,29,31,27)(26,35,32,33)(28,39,38,36)(30,37,34,40);(1,5,6,2)(3,8,7,4)(9,14)(10,18)(11,22)(12,16)(13,19)(15,24)(17,20)(21,23)(25,30,31,34)(26,28,32,38)(27,37,29,40)(33,39,35,36)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,c^2=g,d^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8625382);