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 | |
4631 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10)(11,16)(12,17)(13,21)(14,15)(18,22)(19,20)(23,24)(25,26)(27,33)(28,30)(29,34)(31,32)(35,40)(36,38)(37,39);(1,2)(3,6)(4,7)(5,8)(10,15)(12,16)(18,21)(20,23)(25,27)(26,30)(28,31)(29,36)(32,33)(34,40)(35,37)(38,39);(10,15)(11,17)(18,21)(19,24)(26,32)(27,28)(29,37)(38,40);(1,3,5,4)(2,6,8,7)(9,11,14,17)(10,16,15,12)(13,19,22,24)(18,23,21,20)(25,28,31,27)(26,33,32,30)(29,36,37,35)(34,40,39,38);(1,3,5,4)(2,6,8,7)(9,12,14,16)(10,17,15,11)(13,20,22,23)(18,24,21,19)(25,29,31,37)(26,34,32,39)(27,35,28,36)(30,38,33,40);(1,4,5,3)(2,7,8,6)(9,13,14,22)(10,18,15,21)(11,19,17,24)(12,20,16,23)(25,30,31,33)(26,27,32,28)(29,38,37,40)(34,35,39,36)Polycyclic presentation:
2^9|a^2=gh,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=di,e^b=eh,f^b=fi,d^c=dh,e^c=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7897013);