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 | #39048 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4537 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)(3,5)(4,7)(6,8)(9,10,15,17)(11,16,20,24)(12,14,19,22)(13,21,23,18)(25,26,30,35)(27,34,37,31)(28,33,29,32)(36,39,38,40);(2,3)(6,7)(9,11,15,20)(10,16,17,24)(12,18,19,21)(13,22,23,14)(25,27)(26,31)(28,36)(29,38)(30,37)(32,39)(33,40)(34,35);(1,2,5,3)(4,6,8,7)(10,17)(11,20)(14,22)(18,21)(25,28,30,29)(26,32,35,33)(27,36,37,38)(31,39,34,40);(1,3,5,2)(4,7,8,6)(9,12,15,19)(10,14,17,22)(11,21,20,18)(13,16,23,24)(25,29,30,28)(26,33,35,32)(27,36,37,38)(31,39,34,40);(1,4)(2,6)(3,7)(5,8)(9,13)(10,18)(11,14)(12,16)(15,23)(17,21)(19,24)(20,22)(25,26)(27,34)(28,32)(29,33)(30,35)(31,37)(36,40)(38,39);(9,14)(10,19)(11,13)(12,17)(15,22)(16,21)(18,24)(20,23)(25,27)(26,34)(28,38)(29,36)(30,37)(31,35)(32,39)(33,40)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=g,d^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8109368);