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 | #46397 | 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:
(1,2,4,6)(3,5,7,8)(11,17)(12,20)(15,18)(16,21)(25,26)(27,34)(28,38)(29,39)(30,40)(31,35)(32,37)(33,36);(3,7)(5,8)(9,10)(11,18)(12,21)(13,14)(15,17)(16,20)(19,23)(22,24)(25,27,31,37)(26,32,35,34)(28,29,36,40)(30,33,39,38);(1,2)(3,5)(4,6)(7,8)(10,14)(15,18)(16,21)(23,24)(25,28,31,36)(26,33,35,38)(27,29,37,40)(30,32,39,34);(9,11)(10,15)(12,22)(13,17)(14,18)(16,24)(19,20)(21,23)(25,27)(26,34)(28,29)(30,33)(31,37)(32,35)(36,40)(38,39);(1,3)(2,5)(4,7)(6,8)(9,12)(10,16)(11,19)(13,20)(14,21)(15,23)(17,22)(18,24)(25,29)(26,30)(27,36)(28,37)(31,40)(32,33)(34,38)(35,39);(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,21)(22,24)(25,30)(26,29)(27,33)(28,34)(31,39)(32,36)(35,40)(37,38)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,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, 8839787);