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 | |
4589 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10,14,15)(11,19,20,16)(12,17,21,24)(13,22,23,18)(25,26,31,33)(27,37,38,32)(28,29,35,34)(30,40,39,36);(2,3)(6,7)(10,15)(12,21)(16,19)(18,22)(25,27)(26,32)(28,39)(29,40)(30,35)(31,38)(33,37)(34,36);(1,3,5,2)(4,7,8,6)(10,15)(11,20)(17,24)(18,22)(26,33)(27,38)(28,35)(36,40);(1,4,5,8)(2,6,3,7)(9,11)(10,16)(12,22)(13,17)(14,20)(15,19)(18,21)(23,24)(25,28,31,35)(26,34,33,29)(27,39,38,30)(32,36,37,40);(9,12)(10,17)(11,18)(13,19)(14,21)(15,24)(16,23)(20,22)(25,29,31,34)(26,35,33,28)(27,36,38,40)(30,32,39,37);(9,13,14,23)(10,18,15,22)(11,17,20,24)(12,16,21,19)(25,30,31,39)(26,36,33,40)(27,35,38,28)(29,32,34,37)Polycyclic presentation:
2^9|a^2=g,c^2=gh,d^2=i,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10075605);