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 | |
4568 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)(11,12)(13,17)(14,15)(16,19)(18,20)(21,22)(23,24)(26,29)(27,28)(33,37)(35,36);(1,2,5,6)(3,7,8,4)(10,15)(12,16)(17,22)(20,23)(25,26)(27,32)(28,34)(29,31)(30,33)(35,39)(36,40)(37,38);(2,6)(4,7)(13,21)(17,22)(18,24)(20,23)(25,27,31,28)(26,32,29,34)(30,36,38,35)(33,40,37,39);(1,3,5,8)(2,7,6,4)(9,11,14,19)(10,16,15,12)(13,18,21,24)(17,23,22,20)(25,28,31,27)(26,32,29,34)(30,36,38,35)(33,39,37,40);(1,3)(2,4)(5,8)(6,7)(9,12,14,16)(10,11,15,19)(13,20,21,23)(17,18,22,24)(25,29,31,26)(27,34,28,32)(30,37,38,33)(35,40,36,39);(1,4)(2,3)(5,7)(6,8)(9,13)(10,17)(11,18)(12,20)(14,21)(15,22)(16,23)(19,24)(25,30)(26,33)(27,35)(28,36)(29,37)(31,38)(32,39)(34,40)Polycyclic presentation:
2^9|b^2=i,c^2=h,d^2=gi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,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, 8265086);