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 | #37824 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4284 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,14)(15,18)(16,20)(22,24)(25,26,31,28)(27,34,33,30)(29,32,38,36)(35,40,39,37);(1,2)(3,7)(4,5)(6,8)(9,10,13,14)(11,15,17,18)(12,20,21,16)(19,24,23,22);(2,5)(6,7)(11,17)(15,18)(19,23)(22,24)(25,27)(26,30)(28,34)(29,35)(31,33)(32,37)(36,40)(38,39);(2,5)(6,7)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,28,31,26)(27,34,33,30)(29,32,38,36)(35,37,39,40);(1,2)(3,6)(4,5)(7,8)(9,11,13,17)(10,15,14,18)(12,19,21,23)(16,22,20,24)(25,29,31,38)(26,32,28,36)(27,35,33,39)(30,37,34,40);(1,3)(2,6)(4,8)(5,7)(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,30,31,34)(26,33,28,27)(29,37,38,40)(32,39,36,35)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10231247);