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 | |
4746 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)(11,18)(12,22)(13,20)(14,16)(15,21)(17,23)(19,24)(25,26)(27,31)(28,33)(29,37)(30,32)(34,39)(35,36)(38,40);(2,3)(6,7)(9,11)(10,15)(12,20)(13,17)(14,21)(16,18)(19,23)(22,24)(25,27,30,33)(26,31,32,28)(29,35,39,40)(34,38,37,36);(1,2,5,3)(4,6,8,7)(10,16)(12,23)(15,18)(19,20)(26,32)(27,33)(34,37)(35,40);(9,12)(10,17)(11,20)(13,15)(14,23)(16,22)(18,24)(19,21)(25,26)(27,31)(28,33)(29,34)(30,32)(35,38)(36,40)(37,39);(1,3,5,2)(4,6,8,7)(9,11)(10,18)(12,20)(13,22)(14,21)(15,16)(17,24)(19,23)(25,28,30,31)(26,33,32,27)(29,38,39,36)(34,35,37,40);(1,4)(2,6)(3,7)(5,8)(9,13,14,24)(10,19,16,20)(11,17,21,22)(12,15,23,18)(25,29)(26,34)(27,35)(28,36)(30,39)(31,38)(32,37)(33,40)Polycyclic presentation:
2^9|b^2=gh,c^2=ghi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9755161);