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:
(1,2,4,5)(3,7,8,6)(9,10,14,15)(11,19,20,16)(12,13,18,17)(21,22,24,23)(25,26,31,30)(27,34,29,32)(28,33,36,38)(35,40,37,39);(2,5)(6,7)(10,15)(11,20)(12,18)(22,23)(28,36)(33,38)(35,37)(39,40);(1,2)(3,6)(4,5)(7,8)(10,15)(12,18)(16,19)(21,24)(25,27,31,29)(26,32,30,34)(28,35,36,37)(33,39,38,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,21)(13,23)(14,20)(15,19)(17,22)(18,24)(25,28)(26,33)(27,35)(29,37)(30,38)(31,36)(32,39)(34,40);(1,3,4,8)(2,6,5,7)(9,12,14,18)(10,17,15,13)(11,21,20,24)(16,22,19,23)(25,29,31,27)(26,34,30,32)(28,37,36,35)(33,40,38,39);(9,13,14,17)(10,18,15,12)(11,22,20,23)(16,21,19,24)(25,30,31,26)(27,34,29,32)(28,38,36,33)(35,40,37,39)Polycyclic presentation:
2^9|a^2=h,c^2=gh,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8541096);