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 | |
4600 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)(4,6)(5,8)(9,10)(11,12)(13,17)(14,15)(16,18)(19,20)(21,22)(23,24)(25,26,31,32)(27,29,28,33)(30,34,38,39)(35,37,36,40);(1,2,4,6)(3,5,8,7)(9,10)(11,18)(12,16)(13,17)(14,15)(19,22)(20,21)(23,24)(26,32)(27,28)(34,39)(35,36);(1,2)(3,5)(4,6)(7,8)(10,15)(11,16)(17,24)(19,21)(25,27,31,28)(26,33,32,29)(30,35,38,36)(34,40,39,37);(1,3,4,8)(2,5,6,7)(9,11,14,16)(10,12,15,18)(13,21,23,19)(17,22,24,20)(25,28,31,27)(26,29,32,33)(30,36,38,35)(34,37,39,40);(9,12)(10,16)(11,15)(13,22)(14,18)(17,19)(20,23)(21,24)(25,29)(26,27)(28,32)(30,37)(31,33)(34,35)(36,39)(38,40);(9,13,14,23)(10,17,15,24)(11,19,16,21)(12,20,18,22)(25,30,31,38)(26,34,32,39)(27,35,28,36)(29,37,33,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,c^2=ghi,d^2=i,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, 8628002);