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 | #38361 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4397 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)(11,21)(12,13)(14,18)(15,19)(16,20)(17,22)(23,24)(25,26,31,29)(27,34,32,28)(30,33,38,37)(35,40,39,36);(1,3,5,8)(2,6,4,7)(9,11,16,22)(10,17,20,21)(12,23,14,19)(13,15,18,24)(25,27)(26,28)(29,34)(30,35)(31,32)(33,36)(37,40)(38,39);(1,2,5,4)(3,6,8,7)(9,12,16,14)(10,13,20,18)(11,19,22,23)(15,17,24,21)(25,27)(26,28)(29,34)(30,35)(31,32)(33,36)(37,40)(38,39);(1,2,5,4)(3,6,8,7)(9,13,16,18)(10,12,20,14)(11,15,22,24)(17,23,21,19)(25,28,31,34)(26,32,29,27)(30,36,38,40)(33,39,37,35);(1,4,5,2)(3,7,8,6)(9,14,16,12)(10,18,20,13)(11,19,22,23)(15,17,24,21)(25,29,31,26)(27,34,32,28)(30,33,38,37)(35,36,39,40);(9,15)(10,19)(11,18)(12,17)(13,22)(14,21)(16,24)(20,23)(25,30)(26,33)(27,35)(28,36)(29,37)(31,38)(32,39)(34,40)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=gh,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^e=fhi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7561437);