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 | |
4517 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,13,16)(11,14,19,17)(12,15,20,22)(18,21,24,23)(25,26)(27,35)(28,30)(29,32)(31,33)(34,39)(36,37)(38,40);(1,2,5,4)(3,6,7,8)(9,10)(11,14)(12,15)(13,16)(17,19)(18,21)(20,22)(23,24)(25,27,31,29)(26,32,33,35)(28,38,39,36)(30,37,34,40);(2,4)(6,8)(12,20)(15,22)(18,24)(21,23)(26,33)(30,34)(32,35)(37,40);(3,7)(6,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,28)(26,34)(27,36)(29,38)(30,33)(31,39)(32,37)(35,40);(1,3)(2,6)(4,8)(5,7)(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,29,31,27)(26,35,33,32)(28,36,39,38)(30,40,34,37);(1,4,5,2)(3,8,7,6)(9,12)(10,15)(11,18)(13,20)(14,21)(16,22)(17,23)(19,24)(25,30,31,34)(26,28,33,39)(27,37,29,40)(32,38,35,36)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8232888);