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 | |
4655 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,11)(15,17)(16,18)(22,23)(25,26)(27,33)(28,32)(29,30)(31,35)(34,40)(36,39)(37,38);(1,2)(3,6)(4,5)(7,8)(9,10,14,11)(12,15,19,17)(13,16,21,18)(20,22,24,23)(25,27)(26,31)(28,36)(29,33)(30,35)(32,40)(34,37)(38,39);(2,5)(3,7)(12,19)(13,21)(15,17)(16,18)(26,29)(28,37)(31,33)(34,36);(3,7)(6,8)(9,11,14,10)(12,15,19,17)(13,18,21,16)(20,22,24,23)(25,27)(26,31)(28,36)(29,33)(30,35)(32,40)(34,37)(38,39);(1,3)(2,6)(4,7)(5,8)(9,12,14,19)(10,15,11,17)(13,20,21,24)(16,22,18,23)(25,28,30,37)(26,32,29,38)(27,34,35,36)(31,39,33,40);(1,2)(3,6)(4,5)(7,8)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,29,30,26)(27,33,35,31)(28,38,37,32)(34,40,36,39)Polycyclic presentation:
2^9|b^2=gh,d^2=gh,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,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, 8843793);