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 | |
4337 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,16)(12,20)(13,17)(14,19)(15,21)(18,22)(23,24)(25,26)(27,35)(28,31)(29,33)(30,38)(32,36)(34,39)(37,40);(2,3)(6,7)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,27,31,36)(26,32,28,35)(29,39,40,30)(33,38,37,34);(1,3,5,2)(4,7,8,6)(26,28)(32,35)(33,37)(34,38);(1,4,5,8)(2,6,3,7)(9,12,13,21)(10,15,17,20)(11,18,19,24)(14,22,16,23)(25,28,31,26)(27,35,36,32)(29,37,40,33)(30,38,39,34);(9,11)(10,16)(12,18)(13,19)(14,17)(15,23)(20,22)(21,24)(25,29,31,40)(26,33,28,37)(27,30,36,39)(32,34,35,38);(9,10)(11,14)(12,15)(13,17)(16,19)(18,22)(20,21)(23,24)(25,30)(26,34)(27,29)(28,38)(31,39)(32,33)(35,37)(36,40)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=i,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9592693);