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 | #55499 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4232 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10)(11,16)(12,18)(13,22)(14,21)(15,17)(19,23)(20,24)(25,26)(27,34)(28,31)(29,36)(30,33)(32,38)(35,37)(39,40);(3,7)(6,8)(9,11)(10,16)(12,15)(13,20)(14,19)(17,18)(21,23)(22,24)(26,28)(27,33)(32,36)(35,39);(1,2)(3,6)(4,5)(7,8)(10,17)(11,12)(19,22)(20,21)(25,27)(26,30)(28,34)(29,35)(31,33)(32,37)(36,40)(38,39);(1,3)(2,6)(4,7)(5,8)(9,12,15,11)(10,18,17,16)(13,20,23,21)(14,22,24,19)(25,28,31,26)(27,34,33,30)(29,36,38,32)(35,40,39,37);(1,2)(3,6)(4,5)(7,8)(9,13,15,23)(10,19,17,22)(11,21,12,20)(14,16,24,18)(25,29)(26,32)(27,35)(28,36)(30,37)(31,38)(33,39)(34,40);(9,14)(10,20)(11,19)(12,22)(13,16)(15,24)(17,21)(18,23)(25,30,31,34)(26,33,28,27)(29,37,38,40)(32,39,36,35)Polycyclic presentation:
2^9|d^2=i,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8021543);