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 | #51815 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4352 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10,14,15)(11,12,19,16)(13,17,21,22)(18,20,24,23)(25,26)(27,29)(28,36)(30,31)(32,34)(33,38)(35,39)(37,40);(2,6)(3,7)(10,15)(12,16)(13,21)(18,24)(26,31)(27,34)(33,36)(35,40);(1,3)(2,4)(5,7)(6,8)(9,11,14,19)(10,12,15,16)(13,18,21,24)(17,20,22,23)(25,27,30,34)(26,32,31,29)(28,35,38,40)(33,39,36,37);(1,4)(2,3)(5,8)(6,7)(9,11)(10,16)(12,15)(13,18)(14,19)(17,23)(20,22)(21,24)(25,28)(26,33)(27,35)(29,37)(30,38)(31,36)(32,39)(34,40);(9,12,14,16)(10,11,15,19)(13,20,21,23)(17,18,22,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40);(9,13)(10,17)(11,18)(12,20)(14,21)(15,22)(16,23)(19,24)(25,29,30,32)(26,34,31,27)(28,37,38,39)(33,40,36,35)Polycyclic presentation:
2^9|a^2=gh,c^2=ghi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8664278);