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 | |
4798 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(11,17)(15,18)(19,23)(22,24)(25,26,30,31)(27,28,35,32)(29,33,37,38)(34,36,40,39);(3,7)(5,8)(9,10)(11,18)(12,20)(13,14)(15,17)(16,21)(19,22)(23,24)(26,31)(28,32)(29,37)(34,40);(1,2)(3,5)(4,6)(7,8)(10,14)(15,18)(16,20)(22,24)(25,27,30,35)(26,28,31,32)(29,34,37,40)(33,36,38,39);(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,27)(26,32)(28,31)(29,34)(30,35)(33,39)(36,38)(37,40);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,24)(21,23)(25,28,30,32)(26,27,31,35)(29,36,37,39)(33,34,38,40);(1,2)(3,5)(4,6)(7,8)(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,29,30,37)(26,33,31,38)(27,34,35,40)(28,36,32,39)Polycyclic presentation:
2^9|a^2=gh,c^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8658963);