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:
(1,2,5,3)(4,7,8,6)(9,10)(11,17)(12,20)(13,14)(15,18)(16,22)(19,23)(21,24)(25,26)(27,30)(28,36)(29,33)(31,35)(32,38)(34,40)(37,39);(2,3)(6,7)(10,14)(15,17)(16,20)(23,24)(25,27,29,35)(26,30,33,31)(28,37,38,34)(32,40,36,39);(1,3,5,2)(4,7,8,6)(11,18)(15,17)(19,21)(23,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,40)(36,39)(37,38);(1,4,5,8)(2,6,3,7)(9,11)(10,15)(12,21)(13,18)(14,17)(16,24)(19,22)(20,23)(25,26)(27,30)(28,32)(29,33)(31,35)(34,39)(36,38)(37,40);(9,12,13,22)(10,16,14,20)(11,19,18,21)(15,23,17,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,40)(36,39)(37,38);(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,22)(21,24)(25,28,29,38)(26,32,33,36)(27,34,35,37)(30,39,31,40)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=hi,d^2=hi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8780817);