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 |
| prime radical | #2428 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4273 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,16)(12,18)(13,14)(15,17)(19,24)(20,21)(22,23)(25,26,32,36)(27,35,30,31)(28,34,29,33)(37,40,38,39);(1,2,5,4)(3,6,7,8)(9,11)(10,16)(12,17)(13,20)(14,21)(15,18)(19,23)(22,24)(25,27,32,30)(26,31,36,35)(28,37,29,38)(33,40,34,39);(2,4)(6,8)(10,17)(11,18)(14,19)(20,22)(25,28,32,29)(26,33,36,34)(27,37,30,38)(31,40,35,39);(3,7)(6,8)(9,12,15,16)(10,18,17,11)(13,23,24,21)(14,22,19,20)(25,29,32,28)(26,34,36,33)(27,37,30,38)(31,40,35,39);(1,3)(2,6)(4,8)(5,7)(9,13)(10,19)(11,20)(12,21)(14,17)(15,24)(16,23)(18,22)(25,30,32,27)(26,35,36,31)(28,38,29,37)(33,39,34,40);(1,4,5,2)(3,8,7,6)(9,14,15,19)(10,13,17,24)(11,21,18,23)(12,22,16,20)(25,31,32,35)(26,27,36,30)(28,39,29,40)(33,38,34,37)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=g,d^2=h,e^2=g,f^2=gi,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, 8232152);