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 | |
4588 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,4)(6,7)(9,10)(11,15)(12,20)(13,21)(14,16)(17,23)(18,24)(19,22)(25,26)(27,33)(28,29)(30,38)(31,32)(34,39)(35,36)(37,40);(1,2,5,8)(3,6,4,7)(9,11,14,19)(10,15,16,22)(12,18,23,21)(13,20,24,17)(25,27,31,29)(26,28,32,33)(30,35,39,37)(34,36,38,40);(1,3,5,4)(2,6,8,7)(10,16)(11,19)(17,20)(18,21)(26,32)(28,33)(34,38)(36,40);(1,2)(3,6)(4,7)(5,8)(9,11)(10,15)(12,21)(13,20)(14,19)(16,22)(17,24)(18,23)(25,28)(26,27)(29,32)(30,36)(31,33)(34,35)(37,38)(39,40);(1,4,5,3)(2,7,8,6)(9,12,14,23)(10,17,16,20)(11,18,19,21)(13,22,24,15)(25,29,31,27)(26,33,32,28)(30,37,39,35)(34,40,38,36);(9,13)(10,18)(11,20)(12,22)(14,24)(15,23)(16,21)(17,19)(25,30,31,39)(26,34,32,38)(27,35,29,37)(28,36,33,40)Polycyclic presentation:
2^9|b^2=h,c^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8064387);