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 | |
4439 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)(4,6)(5,8)(9,10,14,15)(11,19,20,16)(12,22,23,17)(13,18,24,21)(25,26)(27,32)(28,30)(29,33)(31,35)(34,39)(36,37)(38,40);(1,2,4,6)(3,5,8,7)(10,15)(11,20)(17,22)(18,21)(25,27)(26,31)(28,32)(29,34)(30,35)(33,38)(36,39)(37,40);(1,2)(3,5)(4,6)(7,8)(10,15)(13,24)(16,19)(17,22)(29,37)(33,36)(34,40)(38,39);(1,3)(2,5)(4,8)(6,7)(9,11,14,20)(10,16,15,19)(12,18,23,21)(13,17,24,22)(25,28,30,26)(27,32,35,31)(29,36,37,33)(34,39,40,38);(9,12,14,23)(10,17,15,22)(11,21,20,18)(13,19,24,16)(25,27)(26,32)(28,31)(29,34)(30,35)(33,39)(36,38)(37,40);(9,13)(10,18)(11,17)(12,19)(14,24)(15,21)(16,23)(20,22)(25,29)(26,33)(27,34)(28,36)(30,37)(31,38)(32,39)(35,40)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=gi,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8202733);