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 | |
4808 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,5)(4,7)(6,8)(9,10)(11,12)(13,18)(14,15)(16,17)(19,20)(21,24)(22,23)(25,26)(27,33)(28,30)(29,32)(31,35)(34,39)(36,37)(38,40);(2,3)(6,7)(10,15)(11,17)(18,23)(19,21)(25,27)(26,31)(28,33)(29,34)(30,35)(32,38)(36,39)(37,40);(1,2,5,3)(4,6,8,7)(10,15)(12,16)(18,23)(20,24)(29,37)(32,36)(34,40)(38,39);(1,3,5,2)(4,7,8,6)(9,11,14,17)(10,16,15,12)(13,21,22,19)(18,20,23,24)(25,28,30,26)(27,33,35,31)(29,36,37,32)(34,39,40,38);(1,4)(2,6)(3,7)(5,8)(9,12,14,16)(10,17,15,11)(13,20,22,24)(18,21,23,19)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40);(9,13)(10,18)(11,19)(12,20)(14,22)(15,23)(16,24)(17,21)(25,27)(26,33)(28,31)(29,34)(30,35)(32,39)(36,38)(37,40)Polycyclic presentation:
2^9|c^2=i,d^2=g,e^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8441885);