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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4669 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,17)(12,20)(13,15)(14,19)(16,22)(18,23)(21,24)(25,26,30,29)(27,31,35,34)(28,36,37,32)(33,40,38,39);(1,2)(3,7)(4,5)(6,8)(9,11,13,19)(10,14,15,17)(12,21,22,18)(16,24,20,23)(26,29)(28,37)(31,34)(33,38);(2,5)(6,7)(10,15)(14,17)(16,20)(23,24)(25,27,30,35)(26,31,29,34)(28,33,37,38)(32,39,36,40);(2,5)(6,7)(9,11)(10,14)(12,18)(13,19)(15,17)(16,23)(20,24)(21,22)(25,28)(26,32)(27,33)(29,36)(30,37)(31,39)(34,40)(35,38);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,16)(13,15)(17,19)(18,23)(20,22)(21,24)(25,27)(26,31)(28,38)(29,34)(30,35)(32,40)(33,37)(36,39);(1,3)(2,6)(4,8)(5,7)(9,12,13,22)(10,16,15,20)(11,18,19,21)(14,23,17,24)(25,29,30,26)(27,34,35,31)(28,36,37,32)(33,40,38,39)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10228115);