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 | #2617 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4264 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,19)(12,21)(13,18)(14,15)(16,20)(17,23)(22,24)(25,26,30,33)(27,28,35,31)(29,32,37,39)(34,36,40,38);(1,2)(3,6)(4,5)(7,8)(10,15)(11,20)(17,21)(18,22)(25,27,30,35)(26,31,33,28)(29,34,37,40)(32,38,39,36);(2,5)(6,8)(10,15)(13,24)(16,19)(17,21)(25,27)(26,28)(29,34)(30,35)(31,33)(32,36)(37,40)(38,39);(1,3,4,7)(2,6,5,8)(9,11)(10,16)(12,22)(13,21)(14,20)(15,19)(17,24)(18,23)(25,28)(26,27)(29,36)(30,31)(32,34)(33,35)(37,38)(39,40);(1,3)(2,6)(4,7)(5,8)(9,12,14,23)(10,17,15,21)(11,18,20,22)(13,19,24,16)(25,29,30,37)(26,32,33,39)(27,34,35,40)(28,36,31,38);(1,2)(3,6)(4,5)(7,8)(9,13)(10,18)(11,21)(12,19)(14,24)(15,22)(16,23)(17,20)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,d^2=ghi,e^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, 8195540);