Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2972 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)(10,13)(14,16)(18,22)(23,24);(6,8)(11,15)(14,16)(17,18,21,22)(19,24,20,23);(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,21,20)(18,23,22,24);(1,3,4,2)(9,11)(10,14)(12,15)(13,16)(17,18)(19,23)(20,24)(21,22);(9,10)(11,14)(12,13)(15,16)(17,20,21,19)(18,24,22,23)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=f,d^2=i,e^2=f,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dfg,e^b=ef104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 464754);