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 |
| prime radical | #40413 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4511 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,15)(11,17)(16,23)(18,19)(25,26,30,31)(27,28,33,32)(29,37,39,34)(35,40,38,36);(1,2)(3,6)(4,7)(5,8)(9,10)(11,13)(12,16)(14,15)(17,22)(18,20)(19,24)(21,23)(26,31)(28,32)(34,37)(36,40);(1,3)(2,5)(4,8)(6,7)(11,17)(13,22)(18,19)(20,24)(25,27,30,33)(26,32,31,28)(29,38,39,35)(34,36,37,40);(3,8)(5,6)(9,11)(10,13)(12,19)(14,17)(15,22)(16,24)(18,21)(20,23)(25,27)(26,32)(28,31)(29,35)(30,33)(34,40)(36,37)(38,39);(1,3)(2,6)(4,8)(5,7)(9,12,14,21)(10,16,15,23)(11,18,17,19)(13,20,22,24)(25,28,30,32)(26,33,31,27)(29,36,39,40)(34,38,37,35);(1,2)(3,6)(4,7)(5,8)(9,13)(10,11)(12,20)(14,22)(15,17)(16,18)(19,23)(21,24)(25,29,30,39)(26,34,31,37)(27,35,33,38)(28,36,32,40)Polycyclic presentation:
2^9|a^2=h,c^2=h,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8811626);