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 | |
4694 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)(9,10,14,13)(11,17,18,15)(12,20,22,16)(19,23,21,24)(25,26,30,32)(27,28,35,31)(29,37,39,33)(34,40,38,36);(1,2,6,8)(3,5,7,4)(10,13)(15,17)(16,20)(23,24)(25,27,30,35)(26,31,32,28)(29,38,39,34)(33,36,37,40);(1,3,6,7)(2,4,8,5)(11,18)(15,17)(19,21)(23,24)(26,32)(28,31)(33,37)(36,40);(1,4,6,5)(2,3,8,7)(9,11,14,18)(10,15,13,17)(12,21,22,19)(16,24,20,23)(25,28)(26,27)(29,36)(30,31)(32,35)(33,34)(37,38)(39,40);(1,2)(3,4)(5,7)(6,8)(9,12)(10,16)(11,19)(13,20)(14,22)(15,23)(17,24)(18,21)(25,27)(26,31)(28,32)(29,34)(30,35)(33,40)(36,37)(38,39);(1,5,6,4)(2,7,8,3)(9,13,14,10)(11,17,18,15)(12,20,22,16)(19,24,21,23)(25,29,30,39)(26,33,32,37)(27,34,35,38)(28,36,31,40)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=i,d^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8757750);