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 | |
4414 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,14)(11,17,12,16)(13,18,22,21)(19,24,20,23)(26,32)(27,29)(28,37)(36,40);(1,2)(3,7)(4,5)(6,8)(9,11,15,12)(10,16,14,17)(13,20,22,19)(18,24,21,23)(25,26)(27,33)(28,30)(29,39)(31,32)(34,37)(35,40)(36,38);(2,5)(6,7)(9,11,15,12)(10,17,14,16)(13,19,22,20)(18,24,21,23)(25,27,31,29)(26,33,32,39)(28,35,37,38)(30,40,34,36);(9,12,15,11)(10,17,14,16)(13,20,22,19)(18,24,21,23)(25,28)(26,34)(27,35)(29,38)(30,32)(31,37)(33,36)(39,40);(1,3)(2,6)(4,8)(5,7)(9,13)(10,18)(11,19)(12,20)(14,21)(15,22)(16,23)(17,24)(25,29,31,27)(26,33,32,39)(28,38,37,35)(30,40,34,36);(1,2)(3,6)(4,5)(7,8)(9,14,15,10)(11,17,12,16)(13,21,22,18)(19,24,20,23)(25,30,31,34)(26,28,32,37)(27,36,29,40)(33,38,39,35)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=g,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8106400);