Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| 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×#2535 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3502 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)(9,10)(11,13)(12,15)(14,16)(17,18,20,22)(19,21,23,24);(1,2,4,3)(5,6,7,8)(9,11,14,15)(10,12,16,13)(19,23)(21,24);(5,6)(7,8)(9,12,14,13)(10,15,16,11)(17,18)(19,21)(20,22)(23,24);(1,3,4,2)(9,13,14,12)(10,15,16,11)(17,19)(18,21)(20,23)(22,24);(9,10)(11,12)(13,15)(14,16)(17,18)(19,21)(20,22)(23,24)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,c^2=f,d^2=fgi,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dg,e^c=ef104 conjugacy classes — view the full character table.
| maximal subgroup of | #72278602, C2×#579776 |
| maximal normal subgroup of | #72278602, C2×#579776 |
| maximal quotient of | #72278676, C2×#579776, #72299635, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 486972);