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 | |
4858 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,17)(12,16)(13,22)(14,18)(15,21)(19,20)(23,24)(26,31)(27,34)(29,37)(36,39);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(25,26,30,31)(27,28,34,32)(29,33,37,38)(35,36,40,39);(9,12,14,22)(10,13,18,16)(11,19,21,24)(15,20,17,23)(25,27,30,34)(26,32,31,28)(29,35,37,40)(33,39,38,36);(9,12)(10,16)(11,19)(13,18)(14,22)(15,23)(17,20)(21,24)(25,27)(26,28)(29,35)(30,34)(31,32)(33,36)(37,40)(38,39);(1,3)(2,7)(4,6)(5,8)(9,11)(10,17)(12,19)(13,23)(14,21)(15,18)(16,20)(22,24)(25,28)(26,27)(29,36)(30,32)(31,34)(33,35)(37,39)(38,40);(1,4,5,2)(3,7,8,6)(9,13,14,16)(10,12,18,22)(11,20,21,23)(15,19,17,24)(25,29,30,37)(26,33,31,38)(27,35,34,40)(28,36,32,39)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9820491);