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 | #39408 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4395 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)(11,16)(12,18)(13,21)(14,20)(15,17)(19,24)(22,23)(25,26)(27,33)(28,30)(29,32)(31,35)(34,40)(36,38)(37,39);(1,2,5,4)(3,7,8,6)(9,11,15,14)(10,16,17,20)(12,21,22,19)(13,23,24,18)(26,28)(31,33)(32,36)(39,40);(2,4)(6,7)(10,17)(11,14)(18,23)(19,21)(25,27,30,35)(26,31,28,33)(29,37,38,34)(32,40,36,39);(9,12)(10,18)(11,19)(13,20)(14,21)(15,22)(16,24)(17,23)(25,26,30,28)(27,31,35,33)(29,32,38,36)(34,39,37,40);(1,3,5,8)(2,6,4,7)(9,13)(10,19)(11,18)(12,20)(14,23)(15,24)(16,22)(17,21)(25,28,30,26)(27,33,35,31)(29,36,38,32)(34,40,37,39);(1,4,5,2)(3,7,8,6)(9,14,15,11)(10,20,17,16)(12,21,22,19)(13,23,24,18)(25,29,30,38)(26,32,28,36)(27,34,35,37)(31,39,33,40)Polycyclic presentation:
2^9|b^2=i,c^2=g,d^2=g,e^2=hi,f^2=gi,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, 8161565);