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 | |
4709 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(3,7)(9,10)(11,18)(12,13)(14,15)(16,19)(17,23)(20,21)(22,24)(25,26)(27,33)(28,36)(29,30)(31,35)(32,37)(34,39)(38,40);(2,5)(3,7)(10,15)(13,17)(16,18)(21,24)(26,29)(28,37)(31,33)(34,38);(1,2)(3,6)(4,5)(7,8)(11,19)(16,18)(20,22)(21,24)(25,27,30,35)(26,31,29,33)(28,34,37,38)(32,39,36,40);(2,5)(6,8)(9,11)(10,16)(12,22)(13,21)(14,19)(15,18)(17,24)(20,23)(25,27)(26,31)(28,38)(29,33)(30,35)(32,40)(34,37)(36,39);(1,2)(3,6)(4,5)(7,8)(9,12)(10,17)(11,20)(13,15)(14,23)(16,21)(18,24)(19,22)(25,28)(26,32)(27,34)(29,36)(30,37)(31,39)(33,40)(35,38);(1,3)(2,6)(4,7)(5,8)(9,13,14,17)(10,12,15,23)(11,21,19,24)(16,22,18,20)(25,29,30,26)(27,33,35,31)(28,36,37,32)(34,40,38,39)Polycyclic presentation:
2^9|c^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8736365);