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 | |
4601 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)(3,8)(4,5)(6,7)(9,10)(11,19)(12,16)(13,15)(14,18)(17,22)(20,24)(21,23)(25,26,31,35)(27,32,37,39)(28,38,30,33)(29,34,40,36);(1,3,6,5)(2,4,7,8)(9,11,14,13)(10,15,18,19)(12,20,22,21)(16,23,17,24)(25,27,31,37)(26,32,35,39)(28,34,30,36)(29,38,40,33);(1,3,6,5)(2,4,7,8)(9,12,14,22)(10,16,18,17)(11,20,13,21)(15,23,19,24)(25,28,31,30)(26,33,35,38)(27,36,37,34)(29,32,40,39);(2,7)(3,5)(9,12)(10,17)(11,20)(13,21)(14,22)(15,24)(16,18)(19,23)(25,29,31,40)(26,34,35,36)(27,33,37,38)(28,39,30,32);(1,4,6,8)(2,3,7,5)(9,13,14,11)(10,15,18,19)(12,21,22,20)(16,23,17,24)(25,26)(27,32)(28,33)(29,34)(30,38)(31,35)(36,40)(37,39);(1,5,6,3)(2,4,7,8)(9,10)(11,15)(12,16)(13,19)(14,18)(17,22)(20,23)(21,24)(25,30,31,28)(26,33,35,38)(27,36,37,34)(29,39,40,32)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=gi,d^2=i,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, 8627816);