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 | #42977 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4500 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,13,18,16)(12,20,21,17)(19,24,22,23)(25,26,31,32)(27,29,28,35)(30,33,38,39)(34,37,36,40);(1,2,5,4)(3,7,8,6)(10,15)(11,18)(12,21)(23,24)(26,32)(29,35)(33,39)(37,40);(2,4)(6,7)(10,15)(13,16)(17,20)(23,24)(25,27,31,28)(26,29,32,35)(30,36,38,34)(33,40,39,37);(3,8)(6,7)(9,11,14,18)(10,16,15,13)(12,22,21,19)(17,23,20,24)(25,28,31,27)(26,29,32,35)(30,36,38,34)(33,37,39,40);(1,3)(2,6)(4,7)(5,8)(9,12)(10,17)(11,19)(13,23)(14,21)(15,20)(16,24)(18,22)(25,29)(26,28)(27,32)(30,37)(31,35)(33,36)(34,39)(38,40);(1,4,5,2)(3,7,8,6)(9,13,14,16)(10,18,15,11)(12,23,21,24)(17,22,20,19)(25,30,31,38)(26,33,32,39)(27,34,28,36)(29,37,35,40)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=gh,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8281538);