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 | |
4474 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)(3,7)(4,5)(6,8)(9,10,14,15)(11,12,17,16)(13,21,23,18)(19,22,24,20)(25,26,31,33)(27,30,28,32)(29,34,38,40)(35,37,36,39);(2,5)(6,7)(10,15)(12,16)(18,21)(20,22)(25,27,31,28)(26,32,33,30)(29,36,38,35)(34,37,40,39);(1,2)(3,6)(4,5)(7,8)(10,15)(11,17)(18,21)(19,24)(26,33)(27,28)(34,40)(35,36);(1,3,4,8)(2,6,5,7)(9,11)(10,16)(12,15)(13,19)(14,17)(18,22)(20,21)(23,24)(25,28,31,27)(26,30,33,32)(29,36,38,35)(34,37,40,39);(9,12,14,16)(10,17,15,11)(13,22,23,20)(18,19,21,24)(25,29)(26,34)(27,35)(28,36)(30,37)(31,38)(32,39)(33,40);(9,13,14,23)(10,18,15,21)(11,19,17,24)(12,20,16,22)(25,30)(26,27)(28,33)(29,37)(31,32)(34,35)(36,40)(38,39)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,d^2=i,e^2=gi,f^2=gi,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, 10123746);