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 | #42998 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4561 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)(9,10)(11,17)(12,20)(13,14)(15,19)(16,22)(18,23)(21,24)(25,26)(27,32)(28,38)(29,33)(30,34)(31,35)(36,40)(37,39);(2,3)(6,7)(9,10)(11,15)(12,20)(13,14)(16,22)(17,19)(18,24)(21,23)(25,27,31,36)(26,32,35,40)(28,39,33,30)(29,34,38,37);(1,2,5,3)(4,6,8,7)(10,14)(15,17)(16,20)(23,24)(27,36)(30,39)(32,40)(34,37);(1,3,5,2)(4,7,8,6)(9,11,13,19)(10,15,14,17)(12,21,22,18)(16,24,20,23)(25,28)(26,29)(27,30)(31,33)(32,37)(34,40)(35,38)(36,39);(1,4,5,8)(2,6,3,7)(9,12)(10,16)(11,18)(13,22)(14,20)(15,23)(17,24)(19,21)(25,29)(26,33)(27,37)(28,35)(30,40)(31,38)(32,39)(34,36);(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,22)(21,24)(25,30)(26,34)(27,28)(29,40)(31,39)(32,38)(33,36)(35,37)Polycyclic presentation:
2^9|b^2=hi,c^2=g,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8313998);