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 | |
4476 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)(9,10,15,18)(11,14,17,23)(12,13,16,22)(19,20,21,24)(26,31)(28,35)(33,37)(36,40);(1,2,4,6)(3,5,8,7)(9,11)(10,14)(12,19)(13,20)(15,17)(16,21)(18,23)(22,24)(26,31)(27,32)(33,37)(34,39);(1,2)(3,5)(4,6)(7,8)(11,17)(14,23)(19,21)(20,24)(25,26,30,31)(27,28,32,35)(29,37,38,33)(34,40,39,36);(1,3)(2,5)(4,8)(6,7)(9,12)(10,13)(11,19)(14,20)(15,16)(17,21)(18,22)(23,24)(25,27)(26,28)(29,34)(30,32)(31,35)(33,36)(37,40)(38,39);(9,13)(10,16)(11,20)(12,18)(14,21)(15,22)(17,24)(19,23)(25,28)(26,32)(27,31)(29,36)(30,35)(33,39)(34,37)(38,40);(9,14,15,23)(10,17,18,11)(12,20,16,24)(13,21,22,19)(25,29,30,38)(26,33,31,37)(27,34,32,39)(28,36,35,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8230523);