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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4536 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10)(11,14)(12,17)(13,18)(15,19)(16,21)(20,22)(23,24)(25,26)(27,32)(28,34)(29,38)(30,37)(31,33)(35,39)(36,40);(3,7)(6,8)(10,11)(12,19)(16,18)(20,24)(25,27)(26,32)(28,31)(29,36)(30,35)(33,34)(37,39)(38,40);(1,2)(3,6)(4,5)(7,8)(10,11)(15,17)(16,18)(22,23)(26,33)(27,28)(35,38)(36,37);(1,3)(2,6)(4,7)(5,8)(9,11,14,10)(12,17,19,15)(13,18,21,16)(20,23,24,22)(25,28,31,27)(26,34,33,32)(29,36,39,37)(30,38,40,35);(1,2)(3,6)(4,5)(7,8)(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,29,31,39)(26,35,33,38)(27,37,28,36)(30,32,40,34);(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)(25,30)(26,36)(27,35)(28,38)(29,32)(31,40)(33,37)(34,39)Polycyclic presentation:
2^9|d^2=gi,e^2=i,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, 8021539);