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 | |
4283 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,5)(3,8,4,7)(12,20)(15,23)(19,24)(21,22)(25,26)(27,30)(28,33)(29,34)(31,35)(32,37)(36,38)(39,40);(1,3,6,4)(2,7,5,8)(9,10)(11,16)(12,15)(13,17)(14,18)(19,21)(20,23)(22,24)(26,27)(28,35)(32,34)(36,40);(9,11)(10,13)(12,19)(14,17)(15,22)(16,18)(20,24)(21,23)(25,27,30,26)(28,33,35,31)(29,34,37,32)(36,39,40,38);(1,3,6,4)(2,8,5,7)(9,12,14,20)(10,15,18,23)(11,19,17,24)(13,22,16,21)(25,28)(26,31)(27,33)(29,36)(30,35)(32,38)(34,39)(37,40);(1,4,6,3)(2,7,5,8)(9,11)(10,16)(12,19)(13,18)(14,17)(15,21)(20,24)(22,23)(25,29,30,37)(26,32,27,34)(28,36,35,40)(31,38,33,39);(1,5,6,2)(3,8,4,7)(9,13,14,16)(10,17,18,11)(12,21,20,22)(15,19,23,24)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=h,d^2=gi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8595519);