Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C24 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4964 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,12)(13,15)(14,16);(2,5)(6,8)(17,18,21,20)(19,23,24,22);(9,11)(10,14)(12,15)(13,16)(17,19,21,24)(18,22,20,23);(10,15)(11,16)(17,20,21,18)(19,22,24,23);(9,12,13,14)(10,11,15,16);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^c=dhi128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7540821);