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 | |
4482 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)(3,5)(4,6)(7,8)(9,10)(11,13)(12,15)(14,20)(16,18)(17,22)(19,23)(21,24)(25,26)(27,36)(28,30)(29,33)(31,35)(32,37)(34,39)(38,40);(1,3,7,6)(2,4,8,5)(9,11)(10,16)(12,13)(14,19)(15,18)(17,23)(20,21)(22,24)(27,37)(29,38)(32,36)(33,40);(1,4,7,5)(2,3,8,6)(10,12)(11,18)(17,20)(19,24)(25,27,31,37)(26,32,35,36)(28,38,34,29)(30,33,39,40);(2,8)(4,5)(9,12,15,10)(11,13,18,16)(14,20,22,17)(19,21,24,23)(25,28,31,34)(26,30,35,39)(27,29,37,38)(32,40,36,33);(1,5,7,4)(2,3,8,6)(9,13)(10,11)(12,18)(14,21)(15,16)(17,19)(20,24)(22,23)(25,29)(26,33)(27,28)(30,36)(31,38)(32,39)(34,37)(35,40);(1,6,7,3)(2,5,8,4)(9,14,15,22)(10,17,12,20)(11,19,18,24)(13,21,16,23)(25,30)(26,34)(27,33)(28,35)(29,32)(31,39)(36,38)(37,40)Polycyclic presentation:
2^9|b^2=hi,c^2=gi,d^2=i,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10088396);