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 | |
4717 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)(11,19)(12,21)(13,18)(14,16)(15,22)(17,23)(20,24)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,38)(37,40);(2,3)(6,7)(9,11,14,22)(10,15,16,19)(12,18,23,20)(13,21,24,17)(25,26,29,32)(27,30,35,33)(28,31,36,38)(34,37,40,39);(1,2,5,3)(4,6,8,7)(10,16)(13,24)(15,19)(17,21)(28,36)(31,38)(34,40)(37,39);(1,3,5,2)(4,7,8,6)(9,11)(10,15)(12,20)(13,21)(14,22)(16,19)(17,24)(18,23)(25,27)(26,30)(28,34)(29,35)(31,37)(32,33)(36,40)(38,39);(1,4)(2,6)(3,7)(5,8)(9,12,14,23)(10,17,16,21)(11,20,22,18)(13,15,24,19)(25,26)(27,30)(28,31)(29,32)(33,35)(34,37)(36,38)(39,40);(9,13,14,24)(10,18,16,20)(11,21,22,17)(12,15,23,19)(25,28,29,36)(26,31,32,38)(27,34,35,40)(30,37,33,39)Polycyclic presentation:
2^9|b^2=gi,c^2=i,d^2=i,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8177731);