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 | |
4730 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(11,17)(15,22)(18,20)(23,24)(25,26)(27,34)(28,33)(29,30)(31,35)(32,38)(36,40)(37,39);(1,3,4,7)(2,5,6,8)(9,10)(11,15)(12,19)(13,14)(16,21)(17,22)(18,24)(20,23)(25,27,31,38)(26,32,35,34)(28,36,39,29)(30,37,40,33);(1,2)(3,5)(4,6)(7,8)(10,14)(15,22)(16,19)(23,24)(25,28,31,39)(26,33,35,37)(27,36,38,29)(30,34,40,32);(9,11)(10,15)(12,20)(13,17)(14,22)(16,24)(18,21)(19,23)(25,29)(26,30)(27,28)(31,36)(32,37)(33,34)(35,40)(38,39);(1,3)(2,5)(4,7)(6,8)(9,12)(10,16)(11,18)(13,21)(14,19)(15,23)(17,20)(22,24)(25,27)(26,34)(28,36)(29,39)(30,37)(31,38)(32,35)(33,40);(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,22)(18,23)(19,21)(20,24)(25,30)(26,29)(27,37)(28,32)(31,40)(33,38)(34,39)(35,36)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8832103);