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 | |
4460 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,5)(3,7,8,6)(9,10)(11,17)(12,20)(13,15)(14,19)(16,21)(18,22)(23,24)(25,26,31,30)(27,32,37,36)(28,38,33,29)(34,40,39,35);(2,5)(6,7)(9,11)(10,14)(12,18)(13,19)(15,17)(16,22)(20,23)(21,24)(25,27,31,37)(26,32,30,36)(28,34,33,39)(29,35,38,40);(1,2)(3,6)(4,5)(7,8)(10,15)(14,17)(16,20)(22,23)(25,28)(26,29)(27,34)(30,38)(31,33)(32,35)(36,40)(37,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,18)(13,19)(15,17)(16,22)(20,23)(21,24)(25,27)(26,32)(28,34)(29,35)(30,36)(31,37)(33,39)(38,40);(1,3,4,8)(2,6,5,7)(9,12,13,21)(10,16,15,20)(11,18,19,24)(14,22,17,23)(25,29,31,38)(26,33,30,28)(27,35,37,40)(32,39,36,34);(9,10)(11,14)(12,16)(13,15)(17,19)(18,22)(20,21)(23,24)(25,30,31,26)(27,36,37,32)(28,38,33,29)(34,40,39,35)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8087676);