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 | |
4537 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,13)(11,17,19,15)(12,20,22,16)(18,23,21,24)(25,26)(27,32)(28,34)(29,36)(30,35)(31,33)(37,40)(38,39);(1,2)(3,7)(4,5)(6,8)(9,11,14,19)(10,15,13,17)(12,21,22,18)(16,24,20,23)(25,27)(26,32)(28,30)(29,33)(31,36)(34,35)(37,38)(39,40);(2,5)(6,7)(10,13)(15,17)(16,20)(23,24)(26,33)(27,36)(30,37)(34,39);(9,11)(10,15)(12,18)(13,17)(14,19)(16,23)(20,24)(21,22)(25,28)(26,34)(27,37)(29,35)(30,36)(31,38)(32,40)(33,39);(1,3)(2,6)(4,8)(5,7)(9,12,14,22)(10,16,13,20)(11,18,19,21)(15,23,17,24)(25,29)(26,27)(28,35)(30,39)(31,32)(33,36)(34,37)(38,40);(1,2)(3,6)(4,5)(7,8)(9,13,14,10)(11,17,19,15)(12,20,22,16)(18,24,21,23)(25,30,31,37)(26,35,33,40)(27,28,36,38)(29,39,32,34)Polycyclic presentation:
2^9|a^2=g,b^2=g,e^2=g,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, 8106692);