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 | #49117 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4211 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)(11,19)(12,22)(13,16)(14,20)(15,17)(18,23)(21,24)(25,26,30,33)(27,31,35,29)(28,36,38,32)(34,37,40,39);(1,2)(3,6)(4,5)(7,8)(9,11,15,13)(10,16,17,19)(12,20,23,21)(14,22,24,18)(25,27,30,35)(26,29,33,31)(28,34,38,40)(32,37,36,39);(2,5)(6,8)(10,17)(14,24)(16,19)(18,22)(25,27)(26,31)(28,34)(29,33)(30,35)(32,39)(36,37)(38,40);(3,7)(6,8)(9,12,15,23)(10,18,17,22)(11,21,13,20)(14,16,24,19)(25,28,30,38)(26,32,33,36)(27,34,35,40)(29,37,31,39);(1,3)(2,6)(4,7)(5,8)(9,13,15,11)(10,19,17,16)(12,21,23,20)(14,18,24,22);(1,2)(3,6)(4,5)(7,8)(9,14,15,24)(10,20,17,21)(11,22,13,18)(12,16,23,19)(25,29,30,31)(26,27,33,35)(28,37,38,39)(32,34,36,40)Polycyclic presentation:
2^9|a^2=gh,b^2=g,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8180628);