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 |
| prime radical | #57000 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4067 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,21)(22,24);(2,3)(6,7)(9,11)(10,16)(12,19)(13,15)(14,18)(17,23)(20,22)(21,24)(25,26,30,35)(27,33,36,31)(28,32,38,39)(29,34,40,37);(1,3,5,2)(4,7,8,6)(10,14)(11,13)(12,20)(23,24)(25,27,30,36)(26,31,35,33)(28,34,38,37)(29,39,40,32);(1,4)(2,6)(3,7)(5,8)(9,12,15,20)(10,17,14,21)(11,19,13,22)(16,23,18,24)(25,28)(26,32)(27,34)(29,31)(30,38)(33,40)(35,39)(36,37);(9,13,15,11)(10,18,14,16)(12,19,20,22)(17,23,21,24)(25,27)(26,33)(28,37)(29,32)(30,36)(31,35)(34,38)(39,40);(9,14,15,10)(11,18,13,16)(12,21,20,17)(19,24,22,23)(25,29)(26,34)(27,32)(28,31)(30,40)(33,38)(35,37)(36,39)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=hi,d^2=h,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9405529);