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 | |
4524 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,12)(13,21)(14,15)(16,17)(18,22)(19,23)(20,24)(28,35)(32,39)(34,36)(37,40);(2,3)(6,7)(10,15)(12,16)(13,22)(19,24)(25,26)(27,29)(28,32)(30,31)(33,38)(34,37)(35,39)(36,40);(1,3,5,2)(4,7,8,6)(10,15)(11,17)(18,21)(19,24)(26,31)(27,33)(32,39)(34,36);(1,4,5,8)(2,6,3,7)(9,11)(10,16)(12,15)(13,19)(14,17)(18,23)(20,21)(22,24)(25,27,30,33)(26,29,31,38)(28,36,35,34)(32,40,39,37);(9,12,14,16)(10,17,15,11)(13,20,22,23)(18,24,21,19)(25,28,30,35)(26,32,31,39)(27,34,33,36)(29,37,38,40);(9,13,14,22)(10,18,15,21)(11,19,17,24)(12,20,16,23)(25,29)(26,33)(27,31)(28,37)(30,38)(32,36)(34,39)(35,40)Polycyclic presentation:
2^9|a^2=gh,c^2=gh,d^2=i,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8771746);