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 | |
4413 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(12,20)(16,21)(19,22)(23,24)(25,26)(27,34)(28,32)(29,33)(30,35)(31,38)(36,39)(37,40);(9,10)(11,18)(12,21)(13,15)(14,17)(16,20)(19,23)(22,24)(25,27)(26,31)(28,36)(29,39)(30,38)(32,37)(33,40)(34,35);(1,2)(3,5)(4,6)(7,8)(9,11,14,13)(10,15,17,18)(12,19,20,22)(16,23,21,24)(25,28,30,33)(26,32,35,29)(27,36,38,40)(31,37,34,39);(3,7)(5,8)(9,12)(10,16)(11,19)(13,22)(14,20)(15,23)(17,21)(18,24)(25,29,30,32)(26,33,35,28)(27,37,38,39)(31,36,34,40);(1,3)(2,5)(4,7)(6,8)(9,13,14,11)(10,15,17,18)(12,22,20,19)(16,23,21,24)(25,26)(27,31)(28,32)(29,33)(30,35)(34,38)(36,37)(39,40);(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,16)(13,18)(14,17)(19,23)(20,21)(22,24)(25,27)(26,34)(28,36)(29,37)(30,38)(31,35)(32,39)(33,40)Polycyclic presentation:
2^9|a^2=h,c^2=gi,d^2=g,e^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10418299);