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 | |
4444 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,16)(11,13,19,22)(12,17,21,23)(14,18,20,24)(25,26)(27,35)(28,39)(29,40)(30,34)(31,32)(33,36)(37,38);(1,2)(3,4)(5,6)(7,8)(9,11,15,19)(10,13,16,22)(12,20,21,14)(17,24,23,18)(26,32)(27,36)(28,40)(34,38);(2,6)(4,7)(10,16)(11,19)(14,20)(17,23)(25,27,31,36)(26,33,32,35)(28,37,40,30)(29,38,39,34);(9,12)(10,17)(11,14)(13,18)(15,21)(16,23)(19,20)(22,24)(25,28)(26,29)(27,37)(30,36)(31,40)(32,39)(33,38)(34,35);(1,3)(2,7)(4,6)(5,8)(9,13)(10,11)(12,18)(14,17)(15,22)(16,19)(20,23)(21,24)(25,29)(26,28)(27,38)(30,35)(31,39)(32,40)(33,37)(34,36);(1,4,5,7)(2,3,6,8)(9,14)(10,18)(11,12)(13,17)(15,20)(16,24)(19,21)(22,23)(25,30,31,37)(26,34,32,38)(27,28,36,40)(29,35,39,33)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8139672);