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 | |
4445 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)(3,5)(4,8)(6,7)(25,26,30,35)(27,31,38,40)(28,33,37,32)(29,34,36,39);(2,7)(3,8)(9,10,14,15)(11,16,19,24)(12,17,22,21)(13,18,23,20)(25,27,30,38)(26,31,35,40)(28,36,37,29)(32,34,33,39);(1,3,6,8)(2,4,7,5)(10,15)(11,19)(12,22)(13,23)(25,28,30,37)(26,32,35,33)(27,29,38,36)(31,39,40,34);(1,4,6,5)(2,3,7,8)(9,11)(10,16)(12,18)(13,21)(14,19)(15,24)(17,23)(20,22)(25,28)(26,33)(27,36)(29,38)(30,37)(31,39)(32,35)(34,40);(1,2)(3,4)(5,8)(6,7)(9,12)(10,17)(11,20)(13,16)(14,22)(15,21)(18,19)(23,24)(25,29,30,36)(26,34,35,39)(27,37,38,28)(31,32,40,33);(1,5,6,4)(2,8,7,3)(9,13)(10,18)(11,21)(12,16)(14,23)(15,20)(17,19)(22,24)(25,26)(27,31)(28,33)(29,34)(30,35)(32,37)(36,39)(38,40)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=h,d^2=g,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=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, 8884555);