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 | |
4733 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)(9,10,14,15)(11,12,16,19)(13,17,21,22)(18,20,23,24)(25,26)(27,31)(28,35)(29,37)(30,32)(33,38)(34,40)(36,39);(1,3,5,8)(2,6,7,4)(10,15)(12,19)(17,22)(20,24)(25,27,30,28)(26,31,32,35)(29,34,38,36)(33,39,37,40);(1,3)(2,4)(5,8)(6,7)(11,16)(12,19)(13,21)(17,22)(26,32)(27,28)(29,38)(39,40);(1,4)(2,3)(5,6)(7,8)(9,11)(10,12)(13,18)(14,16)(15,19)(17,20)(21,23)(22,24)(25,28,30,27)(26,31,32,35)(29,36,38,34)(33,39,37,40);(9,12)(10,16)(11,15)(13,20)(14,19)(17,23)(18,22)(21,24)(25,26)(27,31)(28,35)(29,33)(30,32)(34,39)(36,40)(37,38);(9,13,14,21)(10,17,15,22)(11,18,16,23)(12,20,19,24)(25,29,30,38)(26,33,32,37)(27,34,28,36)(31,39,35,40)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8964493);