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 | #40486 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4575 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)(11,19)(12,21)(13,18)(14,15)(16,20)(17,22)(23,24)(25,26,31,33)(27,32,29,28)(30,34,39,40)(35,38,37,36);(2,3)(6,7)(10,15)(11,20)(17,21)(18,24)(25,27,31,29)(26,32,33,28)(30,35,39,37)(34,38,40,36);(1,2,5,3)(4,6,8,7)(10,15)(12,22)(16,19)(18,24)(26,33)(27,29)(34,40)(35,37);(1,3,5,2)(4,7,8,6)(9,11,14,20)(10,16,15,19)(12,18,22,24)(13,21,23,17)(25,28,31,32)(26,27,33,29)(30,38,39,36)(34,37,40,35);(1,4,5,8)(2,6,3,7)(9,12)(10,17)(11,18)(13,16)(14,22)(15,21)(19,23)(20,24)(25,29,31,27)(26,32,33,28)(30,37,39,35)(34,38,40,36);(9,13)(10,18)(11,17)(12,16)(14,23)(15,24)(19,22)(20,21)(25,30)(26,34)(27,35)(28,36)(29,37)(31,39)(32,38)(33,40)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=ghi,d^2=h,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8441820);