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 | |
4445 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,8)(3,4,5,7)(9,10,14,18)(11,17,13,15)(12,16,22,20)(19,24,21,23)(25,26)(27,32)(28,30)(29,36)(31,34)(33,37)(35,38)(39,40);(1,3,6,5)(2,7,8,4)(9,11,14,13)(10,15,18,17)(12,19,22,21)(16,23,20,24)(27,34)(31,32)(35,40)(38,39);(1,4)(2,3)(5,8)(6,7)(25,27)(26,31)(28,32)(29,35)(30,34)(33,38)(36,39)(37,40);(2,8)(3,5)(9,10)(11,15)(12,20)(13,17)(14,18)(16,22)(19,24)(21,23)(25,27)(26,32)(28,31)(29,35)(30,34)(33,39)(36,38)(37,40);(1,5,6,3)(2,4,8,7)(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,24)(18,20)(25,28,30,26)(27,32,34,31)(29,36,37,33)(35,39,40,38);(1,4)(2,3)(5,8)(6,7)(9,13,14,11)(10,17,18,15)(12,21,22,19)(16,24,20,23)(25,29,30,37)(26,33,28,36)(27,35,34,40)(31,38,32,39)Polycyclic presentation:
2^9|a^2=g,b^2=g,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8044176);