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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4548 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(4,8)(6,7)(9,10,13,14)(11,17,19,15)(12,20,21,16)(18,22,24,23)(25,26,30,31)(27,29,34,33)(28,36,38,32)(35,39,40,37);(1,2,5,3)(4,6,8,7)(10,14)(15,17)(16,20)(22,23)(25,26,30,31)(27,33,34,29)(28,32,38,36)(35,39,40,37);(2,3)(6,7)(10,14)(15,17)(16,20)(22,23)(26,31)(27,34)(32,36)(35,40);(1,3,5,2)(4,7,8,6)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,27)(26,29)(28,35)(30,34)(31,33)(32,37)(36,39)(38,40);(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,23)(21,24)(25,28,30,38)(26,32,31,36)(27,35,34,40)(29,37,33,39);(1,4)(2,6)(3,7)(5,8)(9,12,13,21)(10,16,14,20)(11,18,19,24)(15,22,17,23)(25,29)(26,27)(28,37)(30,33)(31,34)(32,35)(36,40)(38,39)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8014727);