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 | |
4562 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)(3,7)(4,8)(5,6)(9,10)(11,18)(12,16)(13,21)(14,15)(17,22)(19,20)(23,24)(25,26)(27,32)(28,30)(29,34)(31,33)(35,39)(36,38)(37,40);(1,2,5,6)(3,7,8,4)(9,11,14,16)(10,12,15,18)(13,19,22,23)(17,20,21,24)(25,27)(26,32)(28,29)(30,34)(31,36)(33,38)(35,37)(39,40);(1,3)(2,4)(5,8)(6,7)(10,15)(12,18)(13,22)(19,23)(26,33)(27,36)(29,37)(30,39);(1,4)(2,3)(5,7)(6,8)(9,11)(10,12)(13,19)(14,16)(15,18)(17,20)(21,24)(22,23)(25,28,31,35)(26,30,33,39)(27,37,36,29)(32,40,38,34);(1,2)(3,4)(5,6)(7,8)(9,12)(10,16)(11,15)(13,20)(14,18)(17,23)(19,21)(22,24)(25,29,31,37)(26,34,33,40)(27,28,36,35)(30,38,39,32);(1,4)(2,3)(5,7)(6,8)(9,13)(10,17)(11,19)(12,20)(14,22)(15,21)(16,23)(18,24)(25,30)(26,35)(27,34)(28,33)(29,38)(31,39)(32,37)(36,40)Polycyclic presentation:
2^9|b^2=g,d^2=gh,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8947895);