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 | |
4344 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,4)(5,7)(6,8)(9,10,14,13)(11,17,19,15)(12,16,20,21)(18,23,24,22)(25,26,31,33)(27,30,28,32)(29,34,38,40)(35,37,36,39);(1,3,5,8)(2,6,7,4)(10,13)(15,17)(16,21)(22,23)(25,27,31,28)(26,32,33,30)(29,36,38,35)(34,37,40,39);(2,7)(4,6)(12,20)(16,21)(18,24)(22,23)(26,33)(27,28)(34,40)(35,36);(1,2,5,7)(3,6,8,4)(9,11)(10,15)(12,18)(13,17)(14,19)(16,22)(20,24)(21,23)(25,28,31,27)(26,30,33,32)(29,36,38,35)(34,37,40,39);(1,4,5,6)(2,3,7,8)(9,12)(10,16)(11,18)(13,21)(14,20)(15,22)(17,23)(19,24)(25,29,31,38)(26,34,33,40)(27,35,28,36)(30,37,32,39);(1,3)(2,6)(4,7)(5,8)(9,13,14,10)(11,17,19,15)(12,21,20,16)(18,23,24,22)(25,30)(26,27)(28,33)(29,37)(31,32)(34,35)(36,40)(38,39)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,d^2=h,e^2=h,f^2=gh,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, 9148155);