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 | |
4399 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,17)(11,16,20,15)(12,21,22,13)(18,19,23,24)(25,26)(27,31)(28,32)(29,36)(30,34)(33,40)(35,37)(38,39);(1,2)(3,8)(4,6)(5,7)(9,11)(10,15)(12,18)(13,19)(14,20)(16,17)(21,24)(22,23)(25,27,30,37)(26,31,34,35)(28,33,39,36)(29,32,40,38);(1,3)(2,4)(5,6)(7,8)(12,22)(13,21)(18,23)(19,24)(25,26,30,34)(27,35,37,31)(28,32,39,38)(29,36,40,33);(3,6)(4,8)(9,12)(10,13)(11,18)(14,22)(15,19)(16,24)(17,21)(20,23)(25,26)(27,31)(28,38)(29,33)(30,34)(32,39)(35,37)(36,40);(1,4,5,8)(2,6,7,3)(9,11)(10,16)(12,23)(13,19)(14,20)(15,17)(18,22)(21,24)(25,28,30,39)(26,32,34,38)(27,36,37,33)(29,35,40,31);(1,2)(3,8)(4,6)(5,7)(9,13)(10,12)(11,19)(14,21)(15,18)(16,23)(17,22)(20,24)(25,29)(26,33)(27,32)(28,35)(30,40)(31,39)(34,36)(37,38)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,e^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8819824);