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 | #35777 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4494 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10)(11,15)(12,20)(13,14)(16,21)(17,19)(18,23)(22,24)(25,26,30,34)(27,29,36,31)(28,33,38,32)(35,37,40,39);(3,7)(6,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,27,30,36)(26,31,34,29)(28,35,38,40)(32,37,33,39);(1,2)(3,6)(4,5)(7,8)(10,14)(12,21)(15,17)(18,24)(25,28,30,38)(26,32,34,33)(27,35,36,40)(29,39,31,37);(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,23)(21,24)(25,26)(27,29)(28,32)(30,34)(31,36)(33,38)(35,39)(37,40);(1,3)(2,6)(4,7)(5,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,28)(26,33)(27,35)(29,37)(30,38)(31,39)(32,34)(36,40);(1,2)(3,6)(4,5)(7,8)(9,12)(10,16)(11,18)(13,21)(14,20)(15,22)(17,23)(19,24)(25,29)(26,27)(28,37)(30,31)(32,40)(33,35)(34,36)(38,39)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8947329);