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 | |
4335 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,12)(13,17)(14,15)(16,19)(18,20)(21,24)(22,23)(25,26)(27,33)(28,30)(29,32)(31,35)(34,40)(36,38)(37,39);(2,5)(6,7)(10,15)(12,16)(17,23)(20,21)(25,27)(26,31)(28,33)(29,34)(30,35)(32,39)(36,40)(37,38);(1,2)(3,6)(4,5)(7,8)(26,28)(31,33)(32,36)(39,40);(9,11)(10,16)(12,15)(13,18)(14,19)(17,21)(20,23)(22,24)(25,28,30,26)(27,33,35,31)(29,36,38,32)(34,40,37,39);(1,2)(3,7)(4,5)(6,8)(9,12,14,16)(10,11,15,19)(13,21,22,20)(17,24,23,18)(25,29,30,38)(26,32,28,36)(27,34,35,37)(31,39,33,40);(1,3)(2,6)(4,8)(5,7)(9,13,14,22)(10,17,15,23)(11,18,19,24)(12,20,16,21)(25,27)(26,31)(28,33)(29,37)(30,35)(32,40)(34,38)(36,39)Polycyclic presentation:
2^9|d^2=h,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9688272);