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 | |
4377 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(5,8)(9,10,14,16)(11,17,12,15)(13,18,21,23)(19,24,20,22)(25,26)(27,29)(28,32)(30,31)(33,34)(35,38)(36,40)(37,39);(1,2,4,6)(3,5,7,8)(9,11,14,12)(10,15,16,17)(13,20,21,19)(18,24,23,22)(25,27)(26,29)(28,35)(30,34)(31,33)(32,38)(36,39)(37,40);(1,3)(2,5)(4,7)(6,8)(10,16)(15,17)(18,23)(22,24)(26,31)(27,34)(32,39)(35,40);(9,12,14,11)(10,17,16,15)(13,19,21,20)(18,22,23,24)(25,28)(26,32)(27,35)(29,38)(30,37)(31,39)(33,36)(34,40);(1,2)(3,5)(4,6)(7,8)(9,13,14,21)(10,18,16,23)(11,19,12,20)(15,22,17,24)(25,29,30,33)(26,27,31,34)(28,36,37,38)(32,40,39,35);(1,3)(2,5)(4,7)(6,8)(9,10)(11,15)(12,17)(13,18)(14,16)(19,22)(20,24)(21,23)(25,27)(26,33)(28,35)(29,31)(30,34)(32,36)(37,40)(38,39)Polycyclic presentation:
2^9|a^2=g,b^2=hi,d^2=g,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8807321);