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 | |
4608 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)(11,17)(12,16)(13,18)(14,15)(19,22)(20,24)(21,23)(25,26,30,34)(27,28,32,31)(29,33,38,40)(35,36,39,37);(1,2,5,6)(3,7,8,4)(10,15)(11,18)(12,20)(22,23)(25,27,30,32)(26,31,34,28)(29,35,38,39)(33,37,40,36);(1,3)(2,4)(5,8)(6,7)(10,15)(11,18)(16,24)(19,21);(2,6)(4,7)(9,11)(10,13)(12,21)(14,18)(15,17)(16,22)(19,20)(23,24)(25,27)(26,31)(28,34)(29,35)(30,32)(33,37)(36,40)(38,39);(1,4)(2,3)(5,7)(6,8)(9,12,14,20)(10,16,15,24)(11,19,18,21)(13,23,17,22)(25,28,30,31)(26,32,34,27)(29,37,38,36)(33,35,40,39);(1,3)(2,7)(4,6)(5,8)(9,13)(10,11)(12,22)(14,17)(15,18)(16,21)(19,24)(20,23)(25,29)(26,33)(27,35)(28,36)(30,38)(31,37)(32,39)(34,40)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,e^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, 10132780);