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 | #2433 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4484 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(13,21)(17,24)(19,23)(20,22)(25,26,31,35)(27,29,38,32)(28,34,30,33)(36,40,37,39);(1,2,4,5)(3,6,7,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,27)(26,32)(28,36)(29,35)(30,37)(31,38)(33,40)(34,39);(9,11)(10,12)(13,19)(14,16)(15,18)(17,20)(21,23)(22,24)(25,28,31,30)(26,33,35,34)(27,36,38,37)(29,39,32,40);(1,3,4,7)(2,6,5,8)(9,11)(10,15)(12,18)(13,19)(14,16)(17,22)(20,24)(21,23)(25,29,31,32)(26,27,35,38)(28,39,30,40)(33,36,34,37);(9,12,14,15)(10,16,18,11)(13,20,21,22)(17,23,24,19)(25,27)(26,29)(28,36)(30,37)(31,38)(32,35)(33,39)(34,40);(9,13)(10,17)(11,19)(12,20)(14,21)(15,22)(16,23)(18,24)(25,30,31,28)(26,34,35,33)(27,37,38,36)(29,40,32,39)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=h,d^2=g,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7553503);