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 | #51745 | 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)(3,8)(4,6)(5,7)(9,10)(11,12)(13,21)(14,15)(16,18)(17,23)(19,22)(20,24)(25,26,31,33)(27,36,30,32)(28,34,37,39)(29,35,40,38);(1,3,5,4)(2,6,7,8)(9,11)(10,12)(13,19)(14,18)(15,16)(17,20)(21,22)(23,24)(25,27,31,30)(26,32,33,36)(28,35,37,38)(29,39,40,34);(10,15)(11,18)(17,21)(19,24)(26,33)(28,37)(32,36)(35,38);(9,11)(10,16)(12,15)(13,19)(14,18)(17,22)(20,21)(23,24)(25,28)(26,34)(27,35)(29,32)(30,38)(31,37)(33,39)(36,40);(1,4,5,3)(2,6,7,8)(9,12,14,16)(10,11,15,18)(13,22,23,20)(17,24,21,19)(25,29)(26,35)(27,34)(28,32)(30,39)(31,40)(33,38)(36,37);(1,2)(3,6)(4,8)(5,7)(9,13,14,23)(10,17,15,21)(11,19,18,24)(12,20,16,22)(25,30,31,27)(26,36,33,32)(28,38,37,35)(29,34,40,39)Polycyclic presentation:
2^9|a^2=hi,b^2=g,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,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, 9625786);