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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4562 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,12)(13,21)(14,15)(16,23)(17,19)(18,22)(20,24)(25,26,30,31)(27,29,33,39)(28,32,37,40)(34,38,35,36);(1,3)(2,6)(4,7)(5,8)(10,15)(12,19)(16,21)(20,22)(25,27)(26,29)(28,35)(30,33)(31,39)(32,36)(34,37)(38,40);(10,15)(11,17)(16,21)(18,24)(26,31)(27,33)(32,40)(34,35);(9,11)(10,12)(13,18)(14,17)(15,19)(16,20)(21,22)(23,24)(25,26)(27,29)(28,32)(30,31)(33,39)(34,38)(35,36)(37,40);(1,3)(2,7)(4,6)(5,8)(9,12)(10,11)(13,22)(14,19)(15,17)(16,24)(18,21)(20,23)(25,28,30,37)(26,32,31,40)(27,34,33,35)(29,38,39,36);(1,4,5,2)(3,7,8,6)(9,13,14,23)(10,16,15,21)(11,18,17,24)(12,20,19,22)(25,29)(26,27)(28,36)(30,39)(31,33)(32,35)(34,40)(37,38)Polycyclic presentation:
2^9|a^2=gh,e^2=i,f^2=g,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, 9622661);