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 | |
4500 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(10,11)(15,17)(16,18)(22,23);(3,7)(5,8)(9,10,14,11)(12,15,20,17)(13,16,21,18)(19,22,24,23)(25,26,30,31)(27,32,36,39)(28,29,33,38)(34,40,37,35);(1,2)(3,5)(4,6)(7,8)(13,21)(16,18)(19,24)(22,23)(26,31)(29,38)(32,39)(35,40);(9,11,14,10)(12,15,20,17)(13,18,21,16)(19,22,24,23)(25,27)(26,32)(28,37)(29,35)(30,36)(31,39)(33,34)(38,40);(1,3)(2,5)(4,7)(6,8)(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,28,30,33)(26,29,31,38)(27,34,36,37)(32,40,39,35);(1,2)(3,5)(4,6)(7,8)(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)(25,29)(26,33)(27,35)(28,31)(30,38)(32,34)(36,40)(37,39)Polycyclic presentation:
2^9|a^2=h,b^2=gi,d^2=g,e^2=gi,f^2=g,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, 8807315);