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 | #54983 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4480 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)(3,8,6,4)(25,26)(27,31)(28,33)(29,37)(30,32)(34,38)(35,36)(39,40);(1,3)(2,4)(5,6)(7,8)(9,10)(11,13)(12,17)(14,15)(16,19)(18,20)(21,22)(23,24)(25,27,30,33)(26,31,32,28)(29,35,38,40)(34,39,37,36);(1,4,5,8)(2,6,7,3)(10,15)(13,16)(17,22)(20,23)(26,32)(27,33)(34,37)(35,40);(1,3)(2,4)(5,6)(7,8)(9,11)(10,16)(12,18)(13,15)(14,19)(17,23)(20,22)(21,24)(25,28,30,31)(26,33,32,27)(29,36,38,39)(34,40,37,35);(1,3)(2,4)(5,6)(7,8)(9,12,14,21)(10,17,15,22)(11,18,19,24)(13,20,16,23)(25,29)(26,34)(27,35)(28,36)(30,38)(31,39)(32,37)(33,40);(1,4)(2,3)(5,8)(6,7)(9,13,14,16)(10,11,15,19)(12,20,21,23)(17,18,22,24)(25,26)(27,31)(28,33)(29,34)(30,32)(35,39)(36,40)(37,38)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=g,d^2=gh,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8104436);