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 | #37300 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4471 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,8,4,6)(9,10)(11,15)(12,13)(14,16)(17,21)(18,19)(20,22)(23,24)(25,26)(27,35)(28,33)(29,36)(30,31)(32,37)(34,39)(38,40);(3,4)(6,8)(9,11,14,20)(10,15,16,22)(12,18,17,23)(13,19,21,24)(26,31)(28,38)(32,35)(34,36);(1,3,5,4)(2,6,7,8)(10,16)(12,17)(15,22)(18,23)(25,27,30,37)(26,32,31,35)(28,36,38,34)(29,33,39,40);(3,4)(6,8)(9,12,14,17)(10,13,16,21)(11,18,20,23)(15,19,22,24)(25,27)(26,32)(28,34)(29,33)(30,37)(31,35)(36,38)(39,40);(1,4,5,3)(2,6,7,8)(9,11)(10,15)(12,18)(13,19)(14,20)(16,22)(17,23)(21,24)(25,28,30,38)(26,33,31,40)(27,36,37,34)(29,32,39,35);(1,2)(3,6)(4,8)(5,7)(9,13)(10,17)(11,19)(12,16)(14,21)(15,23)(18,22)(20,24)(25,29,30,39)(26,34,31,36)(27,33,37,40)(28,35,38,32)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,c^2=gi,d^2=ghi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10090279);