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 | |
4423 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,11)(12,15,20,17)(13,18,22,16)(19,24,21,23)(26,31)(27,33)(34,39)(35,40);(1,2)(3,8)(4,6)(5,7)(9,10,14,11)(12,17,20,15)(13,16,22,18)(19,24,21,23)(25,26)(27,32)(28,33)(29,34)(30,31)(35,37)(36,40)(38,39);(1,3,5,4)(2,6,7,8)(10,11)(15,17)(16,18)(23,24);(9,11,14,10)(12,17,20,15)(13,18,22,16)(19,24,21,23)(25,27,30,33)(26,32,31,28)(29,35,38,40)(34,37,39,36);(1,4,5,3)(2,6,7,8)(9,12,14,20)(10,15,11,17)(13,21,22,19)(16,24,18,23)(25,28,30,32)(26,33,31,27)(29,37,38,36)(34,35,39,40);(1,2)(3,6)(4,8)(5,7)(9,13)(10,16)(11,18)(12,19)(14,22)(15,23)(17,24)(20,21)(25,29,30,38)(26,34,31,39)(27,35,33,40)(28,36,32,37)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=hi,d^2=gh,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,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9613916);