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 | |
4594 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)(11,19)(12,22)(13,21)(14,15)(16,20)(17,23)(18,24)(27,33)(31,34)(35,39)(38,40);(2,3)(6,7)(10,15)(13,24)(16,19)(17,22)(25,26)(27,34)(28,36)(29,30)(31,33)(32,37)(35,38)(39,40);(1,3,5,2)(4,7,8,6)(10,15)(11,20)(17,22)(18,21)(26,30)(31,34)(32,36)(38,40);(1,4,5,8)(2,6,3,7)(9,11,14,20)(10,16,15,19)(12,21,23,18)(13,22,24,17)(25,26)(27,31)(28,32)(29,30)(33,34)(35,38)(36,37)(39,40);(9,12)(10,17)(11,21)(13,16)(14,23)(15,22)(18,20)(19,24)(25,27)(26,31)(28,35)(29,33)(30,34)(32,38)(36,40)(37,39);(9,13)(10,18)(11,22)(12,16)(14,24)(15,21)(17,20)(19,23)(25,28)(26,32)(27,35)(29,37)(30,36)(31,38)(33,39)(34,40)Polycyclic presentation:
2^9|a^2=gh,c^2=gh,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8659073);