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 | |
4486 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,12)(13,21)(14,18)(15,16)(17,22)(19,23)(20,24)(25,26,31,34)(27,30,37,32)(28,38,33,29)(35,36,40,39);(3,7)(5,8)(9,11,14,16)(10,12,18,15)(13,19,22,24)(17,20,21,23)(25,27,31,37)(26,32,34,30)(28,35,33,40)(29,36,38,39);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,18)(13,19)(14,16)(17,23)(20,21)(22,24)(25,28,31,33)(26,29,34,38)(27,35,37,40)(30,39,32,36);(1,3)(2,5)(4,7)(6,8)(9,11)(10,12)(13,19)(14,16)(15,18)(17,20)(21,23)(22,24)(25,26)(27,30)(28,29)(31,34)(32,37)(33,38)(35,39)(36,40);(1,2)(3,5)(4,6)(7,8)(9,12)(10,16)(11,18)(13,20)(14,15)(17,24)(19,21)(22,23)(25,29)(26,33)(27,36)(28,34)(30,35)(31,38)(32,40)(37,39);(9,13,14,22)(10,17,18,21)(11,19,16,24)(12,20,15,23)(25,30)(26,27)(28,39)(29,35)(31,32)(33,36)(34,37)(38,40)Polycyclic presentation:
2^9|a^2=hi,b^2=gi,c^2=g,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8059408);