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 | #46369 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4239 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,8)(4,7)(5,6)(9,10)(11,17)(12,19)(13,21)(14,16)(15,22)(18,20)(23,24);(3,4)(7,8)(9,10,14,16)(11,17,19,12)(13,15,22,21)(18,23,24,20)(25,26,31,35)(27,36,38,32)(28,30,34,40)(29,33,37,39);(1,3)(2,7)(4,6)(5,8)(9,11)(10,12)(13,18)(14,19)(15,20)(16,17)(21,23)(22,24)(25,27,31,38)(26,32,35,36)(28,29,34,37)(30,39,40,33);(3,4)(7,8)(9,12)(10,11)(13,20)(14,17)(15,18)(16,19)(21,24)(22,23)(25,28,31,34)(26,30,35,40)(27,37,38,29)(32,33,36,39);(1,4,6,3)(2,7,5,8)(9,10)(11,12)(13,15)(14,16)(17,19)(18,20)(21,22)(23,24)(25,29,31,37)(26,33,35,39)(27,28,38,34)(30,32,40,36);(1,5,6,2)(3,7,4,8)(9,13,14,22)(10,15,16,21)(11,18,19,24)(12,20,17,23)(25,30)(26,34)(27,33)(28,35)(29,32)(31,40)(36,37)(38,39)Polycyclic presentation:
2^9|b^2=ghi,c^2=hi,d^2=hi,e^2=gi,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, 8923838);