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 | |
4662 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10,14,12)(11,15,17,19)(13,20,21,16)(18,24,23,22)(25,26,30,32)(27,33,34,29)(28,31,36,39)(35,40,37,38);(1,3,5,8)(2,6,7,4)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,26)(27,29)(28,31)(30,32)(33,34)(35,38)(36,39)(37,40);(1,3)(2,4)(5,8)(6,7)(10,12)(11,17)(13,21)(22,24)(27,34)(28,36)(29,33)(31,39);(1,4)(2,3)(5,6)(7,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,27)(26,29)(28,37)(30,34)(31,40)(32,33)(35,36)(38,39);(9,12,14,10)(11,15,17,19)(13,20,21,16)(18,22,23,24)(25,28)(26,31)(27,35)(29,38)(30,36)(32,39)(33,40)(34,37);(9,13,14,21)(10,16,12,20)(11,18,17,23)(15,22,19,24)(25,29)(26,27)(28,38)(30,33)(31,35)(32,34)(36,40)(37,39)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8886417);