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 | |
4528 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10,14,17)(11,13,19,15)(12,20,21,16)(18,23,24,22)(26,30)(28,35)(31,36)(34,39);(1,3,5,8)(2,6,7,4)(9,11,14,19)(10,15,17,13)(12,18,21,24)(16,23,20,22)(27,33)(31,36)(34,39)(38,40);(1,3)(2,4)(5,8)(6,7)(12,21)(16,20)(18,24)(22,23)(25,26,29,30)(27,31,33,36)(28,32,35,37)(34,38,39,40);(1,4)(2,3)(5,6)(7,8)(9,12,14,21)(10,16,17,20)(11,18,19,24)(13,22,15,23)(25,27)(26,31)(28,34)(29,33)(30,36)(32,38)(35,39)(37,40);(9,11)(10,13)(12,18)(14,19)(15,17)(16,22)(20,23)(21,24)(25,28)(26,32)(27,34)(29,35)(30,37)(31,38)(33,39)(36,40);(9,13)(10,11)(12,22)(14,15)(16,18)(17,19)(20,24)(21,23)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=h,d^2=gi,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, 8019728);