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 | |
4396 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)(12,19)(16,20)(18,23)(21,24)(25,26,30,31)(27,34,29,32)(28,36,38,33)(35,39,37,40);(2,6)(3,7)(9,10)(11,13)(12,20)(14,15)(16,19)(17,22)(18,24)(21,23)(26,31)(32,34)(33,36)(39,40);(1,3)(2,4)(5,7)(6,8)(10,15)(11,17)(16,20)(18,23)(25,27,30,29)(26,32,31,34)(28,37,38,35)(33,40,36,39);(1,4)(2,3)(5,8)(6,7)(9,11,14,17)(10,13,15,22)(12,18,19,23)(16,21,20,24)(25,28,30,38)(26,33,31,36)(27,35,29,37)(32,39,34,40);(9,12,14,19)(10,16,15,20)(11,18,17,23)(13,21,22,24)(25,29,30,27)(26,34,31,32)(28,37,38,35)(33,40,36,39);(9,13)(10,17)(11,15)(12,21)(14,22)(16,23)(18,20)(19,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=gi,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8664696);