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 | #45515 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4380 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,15,12)(11,19,22,16)(13,23,21,17)(14,18,20,24)(25,26,31,28)(27,35,37,32)(29,33,40,38)(30,39,36,34);(1,3,5,8)(2,6,4,7)(9,11)(10,16)(12,19)(13,20)(14,21)(15,22)(17,24)(18,23)(25,27,31,37)(26,32,28,35)(29,36,40,30)(33,39,38,34);(10,12)(16,19)(17,23)(18,24)(25,26,31,28)(27,35,37,32)(29,38,40,33)(30,34,36,39);(1,2,5,4)(3,6,8,7)(9,12,15,10)(11,19,22,16)(13,23,21,17)(14,24,20,18)(25,28,31,26)(27,35,37,32)(29,38,40,33)(30,39,36,34);(1,2,5,4)(3,6,8,7)(9,13,15,21)(10,17,12,23)(11,20,22,14)(16,24,19,18)(25,29)(26,33)(27,36)(28,38)(30,37)(31,40)(32,39)(34,35);(1,4,5,2)(3,7,8,6)(9,14,15,20)(10,18,12,24)(11,21,22,13)(16,23,19,17)(25,30,31,36)(26,34,28,39)(27,29,37,40)(32,33,35,38)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=i,d^2=g,e^2=gi,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7618605);