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 | #37439 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4606 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(10,13)(11,17)(16,21)(18,20)(25,26)(27,28)(29,31)(30,37)(32,34)(33,38)(35,40)(36,39);(2,6)(3,7)(9,10,14,13)(11,15,17,19)(12,16,22,21)(18,23,20,24)(25,27,31,32)(26,28,29,34)(30,35,38,39)(33,36,37,40);(1,3)(2,4)(5,7)(6,8)(11,17)(15,19)(18,20)(23,24)(25,28,31,34)(26,32,29,27)(30,36,38,40)(33,39,37,35);(1,4)(2,3)(5,8)(6,7)(9,11)(10,15)(12,20)(13,19)(14,17)(16,24)(18,22)(21,23)(25,28)(26,27)(29,32)(30,36)(31,34)(33,35)(37,39)(38,40);(9,12,14,22)(10,16,13,21)(11,18,17,20)(15,23,19,24)(25,29,31,26)(27,34,32,28)(30,37,38,33)(35,40,39,36);(9,13,14,10)(11,19,17,15)(12,21,22,16)(18,24,20,23)(25,30,31,38)(26,33,29,37)(27,35,32,39)(28,36,34,40)Polycyclic presentation:
2^9|a^2=hi,b^2=gi,c^2=h,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8082925);