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 | #37817 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4290 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,14,11)(12,17,20,15)(13,16,21,18)(19,23,24,22)(25,26,30,33)(27,28,35,31)(29,32,38,40)(34,36,39,37);(1,3,5,8)(2,6,4,7)(25,27,30,35)(26,31,33,28)(29,34,38,39)(32,37,40,36);(1,2,5,4)(3,6,8,7)(10,11)(15,17)(16,18)(22,23)(25,28,30,31)(26,27,33,35)(29,37,38,36)(32,39,40,34);(9,11,14,10)(12,17,20,15)(13,18,21,16)(19,23,24,22)(25,28)(26,27)(29,36)(30,31)(32,34)(33,35)(37,38)(39,40);(1,4,5,2)(3,7,8,6)(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,26)(27,28)(29,32)(30,33)(31,35)(34,36)(37,39)(38,40);(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)(25,29,30,38)(26,32,33,40)(27,34,35,39)(28,36,31,37)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=g,d^2=hi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,e^b=ei,d^c=dh,f^c=fi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7657044);