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 | |
4464 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)(10,11)(15,17)(16,18)(23,24)(25,26)(27,31)(28,32)(29,40)(30,34)(33,37)(35,36)(38,39);(1,3,7,6)(2,4,8,5)(9,10,14,11)(12,15,20,17)(13,16,22,18)(19,23,21,24)(25,27,30,36)(26,31,34,35)(28,37,39,29)(32,33,38,40);(1,4,7,5)(2,3,8,6)(25,26,30,34)(27,35,36,31)(28,32,39,38)(29,33,37,40);(2,8)(4,5)(9,11,14,10)(12,15,20,17)(13,18,22,16)(19,23,21,24)(25,26)(27,31)(28,38)(29,33)(30,34)(32,39)(35,36)(37,40);(1,5,7,4)(2,3,8,6)(9,12,14,20)(10,15,11,17)(13,21,22,19)(16,24,18,23)(25,28,30,39)(26,32,34,38)(27,29,36,37)(31,40,35,33);(1,6,7,3)(2,5,8,4)(9,13,14,22)(10,16,11,18)(12,19,20,21)(15,23,17,24)(25,29)(26,33)(27,28)(30,37)(31,38)(32,35)(34,40)(36,39)Polycyclic presentation:
2^9|b^2=gi,c^2=h,d^2=ghi,e^2=gi,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, 10079025);