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 | |
4349 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)(3,6)(4,5)(7,8)(25,26,30,31)(27,28,33,32)(29,37,39,34)(35,40,38,36);(1,3,5,8)(2,6,4,7)(9,10,14,15)(11,16,21,24)(12,17,22,20)(13,18,23,19)(26,31)(28,32)(34,37)(36,40);(1,3)(2,7)(4,6)(5,8)(10,15)(13,23)(16,24)(17,20)(25,27,30,33)(26,32,31,28)(29,38,39,35)(34,36,37,40);(1,3,5,8)(2,6,4,7)(9,11)(10,16)(12,18)(13,20)(14,21)(15,24)(17,23)(19,22)(25,27)(26,32)(28,31)(29,35)(30,33)(34,40)(36,37)(38,39);(1,3)(2,6)(4,7)(5,8)(9,12)(10,17)(11,19)(13,16)(14,22)(15,20)(18,21)(23,24)(25,28,30,32)(26,33,31,27)(29,36,39,40)(34,38,37,35);(1,4,5,2)(3,7,8,6)(9,13)(10,18)(11,20)(12,16)(14,23)(15,19)(17,21)(22,24)(25,29,30,39)(26,34,31,37)(27,35,33,38)(28,36,32,40)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=g,d^2=gh,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9348151);