Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2962 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,3)(5,6,7,8)(9,10,15,16)(11,12,14,13)(17,18)(19,24)(20,22)(21,23);(5,6,7,8)(9,11,15,14)(10,13,16,12)(17,19)(18,20)(21,22)(23,24);(5,6)(7,8)(9,12,15,13)(10,11,16,14)(17,18)(19,24)(20,22)(21,23);(1,3,4,2)(9,13,15,12)(10,11,16,14)(17,20,21,24)(18,22,23,19);(9,14,15,11)(10,13,16,12)(17,18)(19,20)(21,23)(22,24)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=fgh,d^2=fgi,e^2=fgh,b^a=bf,c^a=cg,c^b=ch,e^d=ef128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 425875);