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 | |
4526 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)(26,31)(27,34)(33,36)(35,38);(1,2,6,8)(3,5,7,4)(9,10)(11,17)(12,20)(13,14)(15,18)(16,21)(19,23)(22,24)(25,26,30,31)(27,29,34,32)(28,36,37,33)(35,40,38,39);(1,3,6,7)(2,4,8,5)(10,14)(11,18)(12,21)(23,24)(27,34)(28,37)(29,32)(33,36);(1,4,6,5)(2,3,8,7)(9,11)(10,15)(12,22)(13,18)(14,17)(16,24)(19,21)(20,23)(25,27,30,34)(26,32,31,29)(28,38,37,35)(33,39,36,40);(1,2)(3,4)(5,7)(6,8)(9,12)(10,16)(11,19)(13,21)(14,20)(15,23)(17,24)(18,22)(25,28)(26,33)(27,35)(29,39)(30,37)(31,36)(32,40)(34,38);(1,5,6,4)(2,7,8,3)(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,21)(22,24)(25,29,30,32)(26,34,31,27)(28,39,37,40)(33,38,36,35)Polycyclic presentation:
2^9|b^2=g,c^2=h,d^2=g,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, 8965378);