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 | |
4574 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)(25,26,31,35)(27,36,38,32)(28,30,39,33)(29,34,40,37);(1,3)(2,5)(4,7)(6,8)(9,10)(11,17)(12,20)(13,14)(15,18)(16,21)(19,23)(22,24)(25,27,31,38)(26,32,35,36)(28,34,39,37)(29,33,40,30);(2,6)(3,7)(10,14)(11,18)(12,21)(23,24)(25,28,31,39)(26,33,35,30)(27,34,38,37)(29,32,40,36);(1,3,4,7)(2,5,6,8)(9,11)(10,15)(12,22)(13,18)(14,17)(16,24)(19,21)(20,23)(25,28)(26,33)(27,37)(29,32)(30,35)(31,39)(34,38)(36,40);(1,3)(2,5)(4,7)(6,8)(9,12)(10,16)(11,19)(13,21)(14,20)(15,23)(17,24)(18,22)(25,29)(26,34)(27,33)(28,36)(30,38)(31,40)(32,39)(35,37);(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,21)(22,24)(25,30,31,33)(26,28,35,39)(27,29,38,40)(32,34,36,37)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8965623);