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 | #36235 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4458 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10,15,17)(11,16,14,18)(12,13,21,24)(19,20,22,23)(25,26)(27,35)(28,31)(29,38)(30,34)(32,36)(33,40)(37,39);(1,2,5,6)(3,7,4,8)(9,11,15,14)(10,16,17,18)(12,19,21,22)(13,20,24,23)(25,27)(26,32)(28,35)(29,37)(30,40)(31,36)(33,39)(34,38);(1,3,5,4)(2,7,6,8)(10,17)(12,21)(16,18)(19,22)(26,28)(27,36)(29,40)(34,39);(9,12)(10,13)(11,19)(14,22)(15,21)(16,20)(17,24)(18,23)(25,28,31,26)(27,35,36,32)(29,38,40,33)(30,39,37,34);(1,3,5,4)(2,7,6,8)(9,13)(10,12)(11,20)(14,23)(15,24)(16,19)(17,21)(18,22)(25,29,31,40)(26,33,28,38)(27,37,36,30)(32,39,35,34);(1,4,5,3)(2,8,6,7)(9,14,15,11)(10,18,17,16)(12,22,21,19)(13,23,24,20)(25,30,31,37)(26,34,28,39)(27,29,36,40)(32,33,35,38)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=gi,d^2=i,e^2=g,f^2=ghi,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, 7894457);