Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #141745 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1972 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #33748 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
339 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 339 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,7,5,8,3,6)(10,14,16,21)(11,17)(12,19)(15,20,23,22)(18,24);(2,6,8,7)(10,15,16,23)(11,18)(14,20,21,22)(17,24);(9,10,11,14,12,15,17,22,13,16,18,21,19,23,24,20)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=e,d^2=g,e^2=h,f^2=g,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=ch,e^b=ei,f^c=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72213975, #72214585, #72214592, … |
| maximal normal subgroup of | #72213975, #72214585, #72214592, … |
| maximal quotient of | #72216172, C3×#141800, #72226064, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48996);