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 | |
4524 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(10,11)(13,21)(15,17)(19,24)(25,26)(27,28)(29,37)(30,34)(31,32)(33,38)(35,39)(36,40);(3,7)(5,8)(9,10,14,11)(12,15,20,17)(13,16,21,18)(19,22,24,23)(25,27,30,32)(26,28,34,31)(29,35,38,40)(33,36,37,39);(1,2)(3,5)(4,6)(7,8)(25,27)(26,31)(28,34)(29,35)(30,32)(33,39)(36,37)(38,40);(1,3)(2,5)(4,7)(6,8)(9,11,14,10)(12,15,20,17)(13,18,21,16)(19,22,24,23)(25,27)(26,28)(29,35)(30,32)(31,34)(33,36)(37,39)(38,40);(1,2)(3,5)(4,6)(7,8)(9,12)(10,15)(11,17)(13,19)(14,20)(16,22)(18,23)(21,24)(25,28)(26,32)(27,34)(29,36)(30,31)(33,40)(35,37)(38,39);(9,13)(10,16)(11,18)(12,19)(14,21)(15,22)(17,23)(20,24)(25,29,30,38)(26,33,34,37)(27,35,32,40)(28,36,31,39)Polycyclic presentation:
2^9|a^2=hi,b^2=gi,d^2=g,f^2=i,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, 8059410);