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 | |
4570 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)(4,5)(6,8)(25,26,30,34)(27,31,32,28)(29,33,39,40)(35,37,38,36);(2,5)(6,7)(9,10,14,15)(11,13,17,22)(12,16,20,23)(18,21,24,19)(25,27)(26,31)(28,34)(29,35)(30,32)(33,37)(36,40)(38,39);(1,2)(3,6)(4,5)(7,8)(10,15)(11,17)(16,23)(18,24)(25,28,30,31)(26,32,34,27)(29,36,39,37)(33,38,40,35);(1,3,4,8)(2,6,5,7)(9,11)(10,13)(12,18)(14,17)(15,22)(16,21)(19,23)(20,24)(25,28)(26,27)(29,37)(30,31)(32,34)(33,38)(35,40)(36,39);(9,12)(10,16)(11,18)(13,21)(14,20)(15,23)(17,24)(19,22)(25,27)(26,28)(29,38)(30,32)(31,34)(33,37)(35,39)(36,40);(9,13)(10,17)(11,15)(12,19)(14,22)(16,18)(20,21)(23,24)(25,29)(26,33)(27,35)(28,36)(30,39)(31,37)(32,38)(34,40)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=h,d^2=g,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, 8616474);