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 | |
4879 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,13,17,21)(12,16,19,22)(18,20,23,24)(25,26,30,34)(27,35,37,31)(28,38,39,32)(29,33,36,40);(1,2)(3,7)(4,5)(6,8)(12,19)(16,22)(18,23)(20,24)(25,27,30,37)(26,31,34,35)(28,29,39,36)(32,33,38,40);(2,5)(6,7)(10,15)(11,17)(16,22)(18,23)(25,28,30,39)(26,32,34,38)(27,36,37,29)(31,40,35,33);(9,11)(10,13)(12,18)(14,17)(15,21)(16,20)(19,23)(22,24)(25,28)(26,32)(27,29)(30,39)(31,33)(34,38)(35,40)(36,37);(1,2)(3,6)(4,5)(7,8)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,29)(26,33)(27,28)(30,36)(31,32)(34,40)(35,38)(37,39);(1,3)(2,6)(4,8)(5,7)(9,13)(10,11)(12,20)(14,21)(15,17)(16,18)(19,24)(22,23)(25,26)(27,31)(28,32)(29,33)(30,34)(35,37)(36,40)(38,39)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9144096);