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 | #2404 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4336 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,12)(13,21)(14,15)(16,23)(17,20)(18,24)(19,22)(25,26,31,32)(27,35,39,33)(28,34,29,30)(36,40,37,38);(1,2)(3,4)(5,6)(7,8)(9,11,14,17)(10,12,15,20)(13,22,23,18)(16,24,21,19)(26,32)(28,29)(33,35)(36,37);(2,6)(4,7)(10,15)(11,17)(16,21)(18,22)(25,27,31,39)(26,33,32,35)(28,36,29,37)(30,40,34,38);(9,12)(10,11)(13,19)(14,20)(15,17)(16,18)(21,22)(23,24)(25,28,31,29)(26,30,32,34)(27,36,39,37)(33,40,35,38);(1,3)(2,7)(4,6)(5,8)(9,13)(10,16)(11,18)(12,19)(14,23)(15,21)(17,22)(20,24)(25,29,31,28)(26,34,32,30)(27,37,39,36)(33,38,35,40);(1,4,5,7)(2,3,6,8)(9,10)(11,12)(13,16)(14,15)(17,20)(18,19)(21,23)(22,24)(25,30,31,34)(26,28,32,29)(27,38,39,40)(33,37,35,36)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=g,d^2=g,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8139600);