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 | |
4488 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(4,8)(6,7)(9,10)(11,14)(12,17)(13,16)(15,20)(18,21)(19,23)(22,24)(25,26)(27,32)(28,37)(29,39)(30,31)(33,38)(34,35)(36,40);(2,3)(6,7)(10,11)(15,17)(16,18)(22,23)(26,31)(32,40)(33,37)(34,39);(1,2,5,3)(4,6,8,7)(9,10,14,11)(12,15,20,17)(13,16,21,18)(19,22,24,23)(25,27,30,36)(26,32,31,40)(28,35,38,29)(33,39,37,34);(9,11,14,10)(12,17,20,15)(13,18,21,16)(19,23,24,22)(25,28)(26,33)(27,35)(29,36)(30,38)(31,37)(32,39)(34,40);(1,3,5,2)(4,7,8,6)(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,40)(37,38);(1,4)(2,6)(3,7)(5,8)(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)(25,29,30,35)(26,34,31,39)(27,28,36,38)(32,33,40,37)Polycyclic presentation:
2^9|c^2=ghi,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8017172);