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 | #55019 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4257 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,14)(12,19)(13,15)(16,20)(17,23)(18,21)(22,24)(25,26,31,36)(27,30,35,32)(28,37,40,33)(29,34,38,39);(1,2,5,6)(3,7,8,4)(9,11,13,18)(10,14,15,21)(12,17,20,24)(16,22,19,23)(25,27,31,35)(26,32,36,30)(28,38,40,29)(33,39,37,34);(1,3)(2,4)(5,8)(6,7)(10,15)(14,21)(16,19)(22,23);(2,6)(4,7)(9,10)(11,14)(12,16)(13,15)(17,22)(18,21)(19,20)(23,24)(25,28,31,40)(26,33,36,37)(27,29,35,38)(30,39,32,34);(1,4)(2,3)(5,7)(6,8)(9,11)(10,14)(12,17)(13,18)(15,21)(16,22)(19,23)(20,24)(25,29)(26,34)(27,28)(30,37)(31,38)(32,33)(35,40)(36,39);(1,3)(2,7)(4,6)(5,8)(9,12)(10,16)(11,17)(13,20)(14,22)(15,19)(18,24)(21,23)(25,30,31,32)(26,35,36,27)(28,39,40,34)(29,33,38,37)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10132746);