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 | #43002 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4615 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)(3,5)(4,7)(6,8)(12,20)(16,21)(18,24)(22,23)(25,26)(27,32)(28,29)(30,34)(31,35)(33,39)(36,40)(37,38);(2,3)(6,7)(9,10)(11,15)(12,21)(13,17)(14,19)(16,20)(18,22)(23,24)(25,27,31,37)(26,32,35,38)(28,36,39,34)(29,40,33,30);(1,2,5,3)(4,6,8,7)(9,11)(10,14)(12,18)(13,19)(15,17)(16,22)(20,24)(21,23)(25,28)(26,33)(27,34)(29,35)(30,38)(31,39)(32,40)(36,37);(1,3,5,2)(4,7,8,6)(9,11)(10,15)(12,18)(13,19)(14,17)(16,23)(20,24)(21,22)(25,29,31,33)(26,28,35,39)(27,30,37,40)(32,34,38,36);(1,4)(2,6)(3,7)(5,8)(9,12)(10,16)(11,18)(13,20)(14,22)(15,23)(17,21)(19,24)(25,30)(26,34)(27,33)(28,38)(29,37)(31,40)(32,39)(35,36);(9,10)(11,14)(12,16)(13,17)(15,19)(18,22)(20,21)(23,24)(25,28)(26,29)(27,36)(30,38)(31,39)(32,40)(33,35)(34,37)Polycyclic presentation:
2^9|b^2=gh,c^2=g,d^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8379027);