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 | #41419 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4170 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10)(11,18)(12,17)(13,15)(14,16)(19,24)(20,22)(21,23)(25,26)(27,34)(28,31)(29,32)(30,33)(35,39)(36,37)(38,40);(1,3,5,8)(2,6,7,4)(9,11,14,13)(10,15,16,18)(12,20,21,19)(17,24,23,22)(25,27,30,31)(26,28,33,34)(29,37,38,35)(32,39,40,36);(1,3)(2,4)(5,8)(6,7)(10,16)(12,21)(15,18)(19,20)(25,28)(26,31)(27,33)(29,36)(30,34)(32,37)(35,40)(38,39);(1,4)(2,3)(5,6)(7,8)(9,12)(10,17)(11,19)(13,20)(14,21)(15,22)(16,23)(18,24)(25,29,30,38)(26,32,33,40)(27,35,31,37)(28,36,34,39);(9,10)(11,18)(12,17)(13,15)(14,16)(19,24)(20,22)(21,23)(25,26)(27,28)(29,32)(30,33)(31,34)(35,36)(37,39)(38,40);(9,13,14,11)(10,18,16,15)(12,20,21,19)(17,24,23,22)(25,28)(26,31)(27,33)(29,36)(30,34)(32,37)(35,40)(38,39)Polycyclic presentation:
2^9|a^2=i,b^2=g,d^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9947772);