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 | |
4394 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(4,8)(25,26,31,30)(27,33,28,32)(29,34,38,37)(35,40,36,39);(2,6)(3,7)(9,10,14,15)(11,13,17,22)(12,16,19,23)(18,21,20,24)(25,27,31,28)(26,32,30,33)(29,35,38,36)(34,39,37,40);(1,2,5,6)(3,8,7,4)(10,15)(12,19)(13,22)(18,20)(25,27,31,28)(26,33,30,32)(29,36,38,35)(34,39,37,40);(2,6)(3,7)(9,11,14,17)(10,13,15,22)(12,20,19,18)(16,24,23,21)(25,28,31,27)(26,33,30,32)(29,35,38,36)(34,39,37,40);(1,3,5,7)(2,4,6,8)(9,12)(10,16)(11,18)(13,21)(14,19)(15,23)(17,20)(22,24)(25,29,31,38)(26,34,30,37)(27,35,28,36)(32,39,33,40);(1,4)(2,7)(3,6)(5,8)(9,13)(10,17)(11,15)(12,21)(14,22)(16,20)(18,23)(19,24)(25,30,31,26)(27,33,28,32)(29,37,38,34)(35,40,36,39)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=gh,d^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9415829);