Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C2×#2867 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4566 conjugacy classes of subgroups; only the 906 normal subgroups are shown.
This group has 4566 conjugacy classes of subgroups — too many to draw. Only its 906 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(5,6,7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,23)(20,21)(22,24);(2,4)(10,11)(12,16)(18,20)(22,23);(1,2,3,4)(17,19)(18,22)(20,23)(21,24);(9,11,13,10)(12,15,16,14)(17,19)(18,22)(20,23)(21,24);(9,12,13,16)(10,14,11,15)(17,20,21,18)(19,23,24,22)Polycyclic presentation:
2^9|a^2=i,c^2=h,d^2=fg,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.
| maximal subgroup of | #72263715, #72263783, #72263792 |
| maximal normal subgroup of | #72263715, #72263783, #72263792 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 464273);