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 | |
4492 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,17,18,12)(13,16,22,23)(19,21,24,20)(25,26,30,32)(27,31,35,38)(28,29,36,37)(33,34,39,40);(1,2,5,6)(3,7,8,4)(10,15)(11,18)(16,23)(19,24)(25,27,30,35)(26,31,32,38)(28,33,36,39)(29,34,37,40);(1,3)(2,4)(5,8)(6,7)(26,32)(28,36)(31,38)(33,39);(2,6)(4,7)(9,11,14,18)(10,12,15,17)(13,19,22,24)(16,20,23,21)(25,28)(26,29)(27,33)(30,36)(31,34)(32,37)(35,39)(38,40);(1,4)(2,3)(5,7)(6,8)(9,12)(10,11)(13,21)(14,17)(15,18)(16,24)(19,23)(20,22)(25,27)(26,31)(28,33)(29,34)(30,35)(32,38)(36,39)(37,40);(1,3)(2,7)(4,6)(5,8)(9,13)(10,16)(11,19)(12,20)(14,22)(15,23)(17,21)(18,24)(25,29)(26,28)(27,34)(30,37)(31,33)(32,36)(35,40)(38,39)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,d^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10132772);