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 | #42448 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4430 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)(11,17)(12,14)(13,16)(15,18)(19,24)(20,22)(21,23)(25,26)(27,35)(28,38)(29,34)(30,32)(31,37)(33,39)(36,40);(1,2)(3,6)(4,7)(5,8)(9,11)(10,15)(12,17)(13,21)(14,18)(16,24)(19,22)(20,23)(25,27,30,37)(26,31,32,35)(28,36,39,29)(33,40,38,34);(1,3,5,4)(2,6,8,7)(10,12)(11,18)(16,20)(19,21)(26,32)(31,35)(33,38)(34,40);(1,3,5,4)(2,6,8,7)(9,12,14,10)(11,17,18,15)(13,20,22,16)(19,24,21,23)(25,26)(27,31)(28,33)(29,34)(30,32)(35,37)(36,40)(38,39);(9,11)(10,15)(12,17)(13,21)(14,18)(16,24)(19,22)(20,23)(25,28)(26,33)(27,36)(29,37)(30,39)(31,40)(32,38)(34,35);(1,4,5,3)(2,7,8,6)(9,13,14,22)(10,16,12,20)(11,19,18,21)(15,23,17,24)(25,29,30,36)(26,34,32,40)(27,28,37,39)(31,33,35,38)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=gh,d^2=ghi,f^2=g,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, 7888409);