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 | #45018 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4455 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,15,16)(11,13,12,19)(14,17,23,24)(18,21,20,22)(25,26,30,35)(27,32,38,31)(28,29,34,33)(36,39,40,37);(3,7)(5,8)(9,11)(10,13)(12,15)(14,18)(16,19)(17,21)(20,23)(22,24)(25,26)(27,31)(28,33)(29,34)(30,35)(32,38)(36,39)(37,40);(1,2)(3,5)(4,6)(7,8)(10,16)(11,12)(17,24)(18,20)(25,27,30,38)(26,31,35,32)(28,36,34,40)(29,37,33,39);(1,3,4,7)(2,5,6,8)(9,12,15,11)(10,13,16,19)(14,18,23,20)(17,22,24,21)(25,27)(26,32)(28,36)(29,39)(30,38)(31,35)(33,37)(34,40);(1,2)(3,5)(4,6)(7,8)(9,13)(10,11)(12,16)(14,22)(15,19)(17,20)(18,24)(21,23)(25,28)(26,33)(27,36)(29,35)(30,34)(31,39)(32,37)(38,40);(9,14,15,23)(10,17,16,24)(11,18,12,20)(13,21,19,22)(25,29,30,33)(26,34,35,28)(27,37,38,39)(31,40,32,36)Polycyclic presentation:
2^9|a^2=h,c^2=g,d^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8653552);