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 |
| prime radical | #42974 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4369 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(6,8)(11,18)(13,21)(15,16)(17,20)(25,26,31,29)(27,33,37,32)(28,38,34,30)(35,36,39,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,14)(13,17)(15,18)(19,23)(20,21)(22,24)(25,27)(26,32)(28,35)(29,33)(30,36)(31,37)(34,39)(38,40);(2,5)(6,8)(9,11,14,18)(10,15,12,16)(13,19,21,24)(17,22,20,23)(25,27)(26,33)(28,39)(29,32)(30,36)(31,37)(34,35)(38,40);(1,2)(3,6)(4,5)(7,8)(9,12,14,10)(11,15,18,16)(13,20,21,17)(19,22,24,23)(25,28)(26,30)(27,35)(29,38)(31,34)(32,36)(33,40)(37,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,15)(13,19)(14,18)(17,23)(20,22)(21,24)(25,29,31,26)(27,33,37,32)(28,38,34,30)(35,40,39,36);(1,3,4,7)(2,6,5,8)(9,13,14,21)(10,17,12,20)(11,19,18,24)(15,22,16,23)(25,30,31,38)(26,34,29,28)(27,36,37,40)(32,39,33,35)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8039638);