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 | |
4523 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)(4,6,8,7)(9,10)(11,19)(12,20)(13,18)(14,16)(15,21)(17,23)(22,24);(1,2)(3,5)(4,6)(7,8)(9,11,14,21)(10,15,16,19)(12,18,23,22)(13,20,24,17)(25,26)(27,33)(28,36)(29,30)(31,35)(32,38)(34,39)(37,40);(4,8)(6,7)(10,16)(13,24)(15,19)(17,20)(26,30)(31,33)(32,36)(39,40);(2,3)(6,7)(9,12)(10,17)(11,18)(13,15)(14,23)(16,20)(19,24)(21,22)(25,27)(26,31)(28,37)(29,35)(30,33)(32,40)(34,38)(36,39);(1,3,5,2)(4,7,8,6)(9,11)(10,15)(12,22)(13,20)(14,21)(16,19)(17,24)(18,23)(25,28)(26,32)(27,34)(29,38)(30,36)(31,39)(33,40)(35,37);(1,4,5,8)(2,6,3,7)(9,13,14,24)(10,18,16,22)(11,20,21,17)(12,15,23,19)(25,26)(27,31)(28,32)(29,30)(33,35)(34,39)(36,38)(37,40)Polycyclic presentation:
2^9|a^2=gh,b^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8292137);