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 | #56607 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4359 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)(9,10,14,17)(11,15,20,23)(12,13,21,16)(18,19,24,22)(25,26)(27,30)(28,29)(31,34)(32,38)(33,39)(35,40)(36,37);(2,4)(6,7)(9,11,14,20)(10,15,17,23)(12,18,21,24)(13,19,16,22)(25,27,31,33)(26,30,34,39)(28,36,38,35)(29,37,32,40);(9,12,14,21)(10,16,17,13)(11,18,20,24)(15,22,23,19)(25,28,31,38)(26,32,34,29)(27,35,33,36)(30,37,39,40);(1,3)(2,6)(4,7)(5,8)(9,12)(10,13)(11,18)(14,21)(15,19)(16,17)(20,24)(22,23)(25,28)(26,29)(27,36)(30,37)(31,38)(32,34)(33,35)(39,40);(1,4,5,2)(3,7,8,6)(9,11)(10,15)(12,18)(13,19)(14,20)(16,22)(17,23)(21,24)(25,29)(26,28)(27,37)(30,36)(31,32)(33,40)(34,38)(35,39);(1,3)(2,7)(4,6)(5,8)(9,13)(10,12)(11,19)(14,16)(15,18)(17,21)(20,22)(23,24)(25,30)(26,33)(27,34)(28,37)(29,35)(31,39)(32,36)(38,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=h,e^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, 9690203);