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 | |
4387 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10)(11,19)(12,16)(13,17)(14,18)(15,20)(21,24)(22,23)(25,26,31,35)(27,36,39,32)(28,30,34,40)(29,38,37,33);(2,6)(3,7)(11,20)(12,22)(15,19)(16,23)(25,27,31,39)(26,32,35,36)(28,37,34,29)(30,38,40,33);(1,3)(2,4)(5,7)(6,8)(9,11,14,20)(10,15,18,19)(12,21,22,13)(16,17,23,24)(25,28,31,34)(26,30,35,40)(27,37,39,29)(32,38,36,33);(1,4)(2,3)(5,8)(6,7)(9,12)(10,16)(11,21)(13,20)(14,22)(15,17)(18,23)(19,24)(25,26)(27,32)(28,30)(29,33)(31,35)(34,40)(36,39)(37,38);(9,13,14,21)(10,17,18,24)(11,12,20,22)(15,23,19,16)(25,29)(26,33)(27,28)(30,32)(31,37)(34,39)(35,38)(36,40);(9,10)(11,15)(12,23)(13,17)(14,18)(16,22)(19,20)(21,24)(25,30)(26,34)(27,38)(28,35)(29,32)(31,40)(33,39)(36,37)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=gi,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8343955);