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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4523 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10,14,17)(11,15,19,13)(12,20,22,16)(18,21,24,23)(25,26)(27,36)(28,33)(29,39)(30,34)(31,35)(32,38)(37,40);(1,2)(3,6)(4,7)(5,8)(25,27)(26,32)(28,30)(29,34)(31,38)(33,40)(35,36)(37,39);(9,11,14,19)(10,13,17,15)(12,18,22,24)(16,21,20,23)(25,28,31,39)(26,29,35,33)(27,37,38,30)(32,40,36,34);(1,3,5,4)(2,6,8,7)(9,11)(10,15)(12,18)(13,17)(14,19)(16,23)(20,21)(22,24)(25,28)(26,33)(27,30)(29,35)(31,39)(32,40)(34,36)(37,38);(9,12,14,22)(10,16,17,20)(11,18,19,24)(13,21,15,23)(25,29,31,33)(26,28,35,39)(27,34,38,40)(30,36,37,32);(1,4,5,3)(2,7,8,6)(9,13,14,15)(10,11,17,19)(12,21,22,23)(16,18,20,24)(25,30)(26,34)(27,28)(29,32)(31,37)(33,36)(35,40)(38,39)Polycyclic presentation:
2^9|a^2=gh,c^2=h,d^2=gi,e^2=h,f^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7777678);