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 | #59392 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4407 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10,15,17)(11,19,13,16)(12,18,22,14)(20,21,23,24)(25,26,30,32)(27,31,34,29)(28,36,37,33)(35,38,40,39);(1,2)(3,6)(4,7)(5,8)(9,11,15,13)(10,16,17,19)(12,20,22,23)(14,21,18,24)(25,27,30,34)(26,31,32,29)(28,35,37,40)(33,39,36,38);(1,3,5,4)(2,6,8,7)(10,17)(12,22)(16,19)(20,23)(26,32)(27,34)(28,37)(38,39);(9,12)(10,18)(11,20)(13,23)(14,17)(15,22)(16,24)(19,21)(25,28)(26,33)(27,35)(29,38)(30,37)(31,39)(32,36)(34,40);(1,4,5,3)(2,7,8,6)(9,13,15,11)(10,19,17,16)(12,23,22,20)(14,24,18,21)(25,26)(27,31)(28,33)(29,34)(30,32)(35,39)(36,37)(38,40);(9,14,15,18)(10,12,17,22)(11,21,13,24)(16,20,19,23)(25,29,30,31)(26,27,32,34)(28,38,37,39)(33,35,36,40)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=ghi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7882230);