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 | #42091 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4549 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10)(11,13)(12,20)(14,15)(16,18)(17,21)(19,22)(23,24)(25,26,31,33)(27,36,39,32)(28,34,40,30)(29,37,38,35);(1,2)(3,6)(4,7)(5,8)(9,11)(10,13)(12,19)(14,18)(15,16)(17,23)(20,22)(21,24)(25,27)(26,32)(28,37)(29,34)(30,38)(31,39)(33,36)(35,40);(1,3,5,4)(2,6,8,7)(10,15)(11,18)(12,21)(22,23)(26,33)(28,40)(32,36)(35,37);(9,11)(10,16)(12,19)(13,15)(14,18)(17,22)(20,23)(21,24)(25,28)(26,34)(27,37)(29,32)(30,33)(31,40)(35,39)(36,38);(1,4,5,3)(2,7,8,6)(9,12)(10,17)(11,19)(13,23)(14,21)(15,20)(16,22)(18,24)(25,29,31,38)(26,35,33,37)(27,34,39,30)(28,32,40,36);(9,13,14,16)(10,11,15,18)(12,22,21,23)(17,24,20,19)(25,30,31,34)(26,28,33,40)(27,38,39,29)(32,37,36,35)Polycyclic presentation:
2^9|a^2=g,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, 7880139);