Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7408 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | AΣL(1,𝔽16) |
| commutator | C2×C4⋊C4 | C22×C4 |
| Frattini, cosocle | #382 | C23 |
| socle | C22 | C2×AΣL(1,𝔽16) |
| prime radical | C24 | #99 |
| LCS | G ▷ C2×C4⋊C4 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C23×C4 ◁ #2285 ◁ G | |
| LECS | G ▷ #382 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
430 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 430 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5,6,7)(10,15,16,23,17,19,21,24)(12,18,20,22);(1,2)(3,7)(4,6)(5,8)(9,10,13,21,14,17,11,16)(12,19,18,24,20,15,22,23);(9,11,14,13)(10,16,17,21)(12,18,20,22)(15,23,19,24)Polycyclic presentation:
2^9|a^2=e,b^2=fh,c^2=i,d^2=gh,e^2=h,f^2=i,b^a=bd,d^a=df,f^a=fh,d^b=dg,e^b=efghi,f^b=fi,g^b=gi,h^b=hi,e^d=ehi,f^d=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72227652 |
| maximal normal subgroup of | #72227652 |
| maximal quotient of | #72227644 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57507);