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 | #2669 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4287 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)(3,4)(5,8)(6,7)(9,10,14,18)(11,16,21,15)(12,13,17,23)(19,22,24,20)(25,26,31,35)(27,34,30,32)(28,33,39,29)(36,40,38,37);(1,3,5,7)(2,4,8,6)(9,11)(10,15)(12,19)(13,20)(14,21)(16,18)(17,24)(22,23)(25,27,31,30)(26,32,35,34)(28,36,39,38)(29,40,33,37);(1,3)(2,6)(4,8)(5,7)(25,28,31,39)(26,29,35,33)(27,36,30,38)(32,40,34,37);(1,2,5,8)(3,4,7,6)(9,12,14,17)(10,13,18,23)(11,19,21,24)(15,20,16,22)(25,28)(26,33)(27,36)(29,35)(30,38)(31,39)(32,37)(34,40);(1,3)(2,4)(5,7)(6,8)(9,11)(10,16)(12,19)(13,22)(14,21)(15,18)(17,24)(20,23)(25,29,31,33)(26,28,35,39)(27,37,30,40)(32,38,34,36);(1,4)(2,7)(3,8)(5,6)(9,13)(10,17)(11,20)(12,18)(14,23)(15,24)(16,19)(21,22)(25,30,31,27)(26,34,35,32)(28,38,39,36)(29,37,33,40)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=hi,d^2=gh,e^2=hi,f^2=hi,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, 9633658);