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 | |
4268 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)(9,10,15,13)(11,18,21,16)(12,17,14,22)(19,24,20,23)(25,26,32,31)(27,35,30,33)(28,38,34,29)(36,37,39,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,19)(13,18)(14,20)(15,21)(17,23)(22,24)(25,27,32,30)(26,33,31,35)(28,39,34,36)(29,40,38,37);(2,5)(6,8)(9,12,15,14)(10,17,13,22)(11,19,21,20)(16,23,18,24)(25,28)(26,29)(27,36)(30,39)(31,38)(32,34)(33,37)(35,40);(9,11)(10,16)(12,19)(13,18)(14,20)(15,21)(17,23)(22,24)(25,29,32,38)(26,34,31,28)(27,37,30,40)(33,39,35,36);(1,2)(3,6)(4,5)(7,8)(9,13,15,10)(11,18,21,16)(12,17,14,22)(19,23,20,24)(25,30,32,27)(26,35,31,33)(28,39,34,36)(29,40,38,37);(1,3,4,7)(2,6,5,8)(9,14,15,12)(10,17,13,22)(11,20,21,19)(16,23,18,24)(25,31,32,26)(27,35,30,33)(28,38,34,29)(36,40,39,37)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=gh,d^2=h,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8030142);