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 | #37940 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4538 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)(9,10,14,16)(11,18,12,15)(13,21,22,17)(19,23,20,24)(26,31)(29,39)(32,38)(33,35);(1,2,5,6)(3,7,4,8)(9,11,14,12)(10,15,16,18)(13,19,22,20)(17,23,21,24)(25,26,30,31)(27,32,36,38)(28,35,37,33)(29,34,39,40);(10,16)(11,12)(17,21)(19,20)(26,31)(28,37)(32,38)(34,40);(1,3,5,4)(2,7,6,8)(9,12,14,11)(10,15,16,18)(13,20,22,19)(17,23,21,24)(25,27,30,36)(26,32,31,38)(28,34,37,40)(29,35,39,33);(1,4,5,3)(2,8,6,7)(9,10)(11,18)(12,15)(13,17)(14,16)(19,24)(20,23)(21,22)(25,28,30,37)(26,33,31,35)(27,34,36,40)(29,38,39,32);(9,13)(10,17)(11,19)(12,20)(14,22)(15,23)(16,21)(18,24)(25,29)(26,34)(27,35)(28,38)(30,39)(31,40)(32,37)(33,36)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=g,e^2=gh,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7623119);