Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| 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 | |
2872 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,3)(5,6,8,7)(9,10)(11,15)(12,14)(13,16)(17,18)(19,23)(20,22)(21,24);(9,10,12,14)(11,13,16,15)(17,18,20,22)(19,21,24,23);(1,2,4,3)(5,7,8,6)(17,18)(19,21)(20,22)(23,24);(1,3,4,2)(9,10)(11,13)(12,14)(15,16)(17,18)(19,21)(20,22)(23,24);(9,11,12,16)(10,13,14,15)(17,19)(18,21)(20,24)(22,23)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gi,d^2=g,e^2=fh,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 430252);