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 |
| prime radical | #43433 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4361 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,4)(5,7)(6,8)(9,10)(11,15)(12,17)(13,21)(14,16)(18,22)(19,24)(20,23);(1,3,5,8)(2,6,7,4)(9,11,14,12)(10,15,16,17)(13,19,22,20)(18,23,21,24)(25,26)(27,33)(28,36)(29,30)(31,34)(32,37)(35,39)(38,40);(1,2,5,7)(3,6,8,4)(10,16)(11,12)(13,22)(23,24)(26,30)(27,34)(28,37)(39,40);(2,7)(3,8)(9,10)(11,17)(12,15)(13,18)(14,16)(19,24)(20,23)(21,22)(25,27)(26,31)(28,38)(29,34)(30,33)(32,40)(35,37)(36,39);(1,2)(3,6)(4,8)(5,7)(9,12,14,11)(10,17,16,15)(13,20,22,19)(18,24,21,23)(25,28)(26,32)(27,35)(29,37)(30,36)(31,39)(33,40)(34,38);(1,4,5,6)(2,8,7,3)(9,13)(10,18)(11,19)(12,20)(14,22)(15,23)(16,21)(17,24)(25,26)(27,31)(28,32)(29,30)(33,34)(35,39)(36,37)(38,40)Polycyclic presentation:
2^9|b^2=h,c^2=g,e^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, 8965984);