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 | |
4703 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,17)(12,20)(13,14)(15,18)(16,21)(19,22)(23,24)(25,26,30,31)(27,32,37,33)(28,34,39,29)(35,36,38,40);(1,2)(3,7)(4,5)(6,8)(10,14)(12,21)(15,17)(19,24)(26,31)(29,34)(32,33)(36,40);(2,5)(6,7)(10,14)(11,18)(16,20)(19,24)(25,27,30,37)(26,32,31,33)(28,38,39,35)(29,36,34,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,23)(21,24)(25,27)(26,33)(28,35)(29,36)(30,37)(31,32)(34,40)(38,39);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,28)(26,34)(27,35)(29,31)(30,39)(32,36)(33,40)(37,38);(1,3)(2,6)(4,8)(5,7)(9,12,13,21)(10,16,14,20)(11,19,18,24)(15,22,17,23)(25,29,30,34)(26,28,31,39)(27,36,37,40)(32,38,33,35)Polycyclic presentation:
2^9|a^2=gh,c^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8665198);