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 | |
4453 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,4)(5,6)(7,8)(9,10,15,12)(11,16,13,18)(14,17,22,20)(19,23,21,24)(25,26)(27,32)(28,30)(29,36)(31,35)(33,37)(34,38)(39,40);(1,3,7,6)(2,4,8,5)(9,11)(10,16)(12,18)(13,15)(14,19)(17,23)(20,24)(21,22)(25,26,30,28)(27,32,35,31)(29,33,37,36)(34,39,40,38);(1,4,7,5)(2,6,8,3)(9,10,15,12)(11,18,13,16)(14,17,22,20)(19,24,21,23)(25,27)(26,31)(28,32)(29,34)(30,35)(33,38)(36,39)(37,40);(1,4,7,5)(2,3,8,6)(9,12,15,10)(11,18,13,16)(14,17,22,20)(19,23,21,24)(25,28,30,26)(27,32,35,31)(29,36,37,33)(34,39,40,38);(1,5,7,4)(2,3,8,6)(9,13,15,11)(10,16,12,18)(14,21,22,19)(17,23,20,24)(25,27)(26,32)(28,31)(29,34)(30,35)(33,39)(36,38)(37,40);(1,6,7,3)(2,4,8,5)(9,14)(10,17)(11,19)(12,20)(13,21)(15,22)(16,23)(18,24)(25,29,30,37)(26,33,28,36)(27,34,35,40)(31,38,32,39)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=gi,d^2=i,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10352139);