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 | #2539 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4307 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)(3,8)(4,7)(5,6)(9,10,14,12)(11,17,18,15)(13,20,21,16)(19,22,24,23);(3,4)(7,8)(10,12)(15,17)(16,20)(22,23)(25,26,30,31)(27,32,35,40)(28,33,38,37)(29,34,39,36);(1,3)(2,7)(4,6)(5,8)(10,12)(11,18)(13,21)(22,23)(26,31)(27,35)(28,38)(29,39);(3,4)(7,8)(9,11,14,18)(10,15,12,17)(13,19,21,24)(16,22,20,23)(25,27)(26,32)(28,34)(29,37)(30,35)(31,40)(33,39)(36,38);(1,4,6,3)(2,7,5,8)(9,12,14,10)(11,15,18,17)(13,20,21,16)(19,22,24,23)(25,28)(26,33)(27,36)(29,32)(30,38)(31,37)(34,35)(39,40);(1,5,6,2)(3,7,4,8)(9,13,14,21)(10,16,12,20)(11,19,18,24)(15,22,17,23)(25,29)(26,34)(27,37)(28,32)(30,39)(31,36)(33,35)(38,40)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,d^2=g,e^2=h,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, 8886363);