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 | #39415 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4298 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,17,18,13)(12,20,21,16)(19,22,24,23)(25,26,31,34)(27,29,33,30)(28,36,39,32)(35,38,40,37);(1,2,5,4)(3,7,8,6)(10,15)(11,18)(12,21)(22,23)(25,27,31,33)(26,30,34,29)(28,35,39,40)(32,38,36,37);(2,4)(6,7)(10,15)(13,17)(16,20)(22,23)(25,27)(26,29)(28,35)(30,34)(31,33)(32,37)(36,38)(39,40);(9,11)(10,13)(12,19)(14,18)(15,17)(16,22)(20,23)(21,24)(25,28,31,39)(26,32,34,36)(27,35,33,40)(29,37,30,38);(1,3,5,8)(2,6,4,7)(9,12,14,21)(10,16,15,20)(11,19,18,24)(13,22,17,23)(25,29,31,30)(26,33,34,27)(28,37,39,38)(32,40,36,35);(1,4,5,2)(3,7,8,6)(9,13)(10,11)(12,22)(14,17)(15,18)(16,19)(20,24)(21,23)(25,30,31,29)(26,27,34,33)(28,38,39,37)(32,35,36,40)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=gh,e^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8161734);