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 | |
4580 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,11)(15,17)(16,18)(22,23)(25,26)(27,32)(28,34)(29,36)(30,35)(31,33)(37,40)(38,39);(1,2)(3,7)(4,5)(6,8)(9,10,14,11)(12,17,20,15)(13,16,21,18)(19,23,24,22)(25,27)(26,32)(28,30)(29,33)(31,36)(34,35)(37,38)(39,40);(2,5)(6,7)(13,21)(16,18)(19,24)(22,23)(26,33)(27,36)(30,37)(34,39);(9,11,14,10)(12,17,20,15)(13,18,21,16)(19,23,24,22)(25,28)(26,34)(27,37)(29,35)(30,36)(31,38)(32,40)(33,39);(1,3,4,8)(2,6,5,7)(9,12)(10,15)(11,17)(13,19)(14,20)(16,22)(18,23)(21,24)(25,29)(26,27)(28,35)(30,39)(31,32)(33,36)(34,37)(38,40);(1,2)(3,6)(4,5)(7,8)(9,13)(10,16)(11,18)(12,19)(14,21)(15,22)(17,23)(20,24)(25,30,31,37)(26,35,33,40)(27,28,36,38)(29,39,32,34)Polycyclic presentation:
2^9|b^2=g,d^2=g,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8122683);