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 |
| prime radical | #35769 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4311 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,18)(11,12,16,15)(13,21,23,17)(19,24,22,20)(25,26)(27,30)(28,33)(29,34)(31,35)(32,38)(36,37)(39,40);(1,2)(3,4)(5,7)(6,8)(9,11)(10,12)(13,19)(14,16)(15,18)(17,20)(21,24)(22,23)(25,27,31,38)(26,32,35,30)(28,36,34,40)(29,37,33,39);(1,3)(2,6)(4,7)(5,8)(9,11,14,16)(10,15,18,12)(13,22,23,19)(17,20,21,24)(25,28,31,34)(26,33,35,29)(27,36,38,40)(30,37,32,39);(3,8)(4,6)(9,11)(10,12)(13,19)(14,16)(15,18)(17,20)(21,24)(22,23)(25,28)(26,29)(27,36)(30,39)(31,34)(32,37)(33,35)(38,40);(1,4)(2,3)(5,6)(7,8)(9,12)(10,16)(11,18)(13,20)(14,15)(17,22)(19,21)(23,24)(25,29)(26,34)(27,37)(28,35)(30,36)(31,33)(32,40)(38,39);(1,2)(3,4)(5,7)(6,8)(9,13,14,23)(10,17,18,21)(11,19,16,22)(12,20,15,24)(25,30)(26,27)(28,39)(29,36)(31,32)(33,40)(34,37)(35,38)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=gh,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, 8877822);