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 | #59009 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4623 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)(11,19)(12,22)(13,17)(14,18)(15,21)(16,23)(20,24)(25,26)(27,30)(28,36)(29,33)(31,35)(32,38)(34,40)(37,39);(1,2)(3,6)(4,7)(5,8)(9,11,14,21)(10,15,18,19)(12,20,23,13)(16,24,22,17)(25,27,29,35)(26,30,33,31)(28,34,38,37)(32,39,36,40);(1,3,5,4)(2,6,8,7)(9,10,14,18)(11,15,21,19)(12,16,23,22)(13,17,20,24)(25,27)(26,31)(28,37)(29,35)(30,33)(32,39)(34,38)(36,40);(9,10)(11,15)(12,16)(13,17)(14,18)(19,21)(20,24)(22,23);(9,12)(10,16)(11,20)(13,21)(14,23)(15,24)(17,19)(18,22)(25,28)(26,32)(27,34)(29,38)(30,39)(31,40)(33,36)(35,37);(1,4,5,3)(2,7,8,6)(9,13,14,20)(10,17,18,24)(11,12,21,23)(15,16,19,22)(25,26)(27,31)(28,32)(29,33)(30,35)(34,40)(36,38)(37,39)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=g,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7641568);