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 | #2424 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4369 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)(11,18)(12,22)(13,17)(14,15)(16,19)(20,24)(21,23)(25,26,31,29)(27,33,28,32)(30,34,38,37)(35,40,36,39);(1,2,5,6)(3,7,8,4)(10,15)(11,19)(12,17)(21,24)(25,27,31,28)(26,32,29,33)(30,35,38,36)(34,39,37,40);(2,6)(4,7)(10,15)(13,22)(16,18)(21,24)(30,38)(34,37)(35,36)(39,40);(1,3,5,8)(2,7,6,4)(9,11)(10,16)(12,23)(13,21)(14,19)(15,18)(17,20)(22,24)(25,28,31,27)(26,33,29,32)(30,36,38,35)(34,40,37,39);(1,3)(2,4)(5,8)(6,7)(9,12,14,17)(10,13,15,22)(11,20,19,23)(16,24,18,21)(25,29,31,26)(27,32,28,33)(30,37,38,34)(35,39,36,40);(1,4)(2,3)(5,7)(6,8)(9,13)(10,17)(11,21)(12,15)(14,22)(16,20)(18,23)(19,24)(25,30)(26,34)(27,35)(28,36)(29,37)(31,38)(32,39)(33,40)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,d^2=ghi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,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, 8265439);