Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10004 |
| automorphism | Aut(G) = C22×#72231762 (order 4096) · Inn(G) = #392 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | #3109 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
283 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 283 conjugacy classes of subgroups — too many to draw. Only its 51 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) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,7)(3,8)(4,5)(9,10)(11,15)(12,14)(13,17)(16,20)(18,23)(19,24)(21,22);(1,2)(3,6)(4,8)(5,7)(10,14)(12,20)(15,23)(16,17)(18,21)(22,24);(1,3,6,2)(4,8,5,7)(9,11,13,19)(10,15,17,24)(12,21,20,18)(14,22,16,23)Polycyclic presentation:
2^8|c^2=g,d^2=g,e^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fh,c^b=cf,d^b=dg,d^c=dh,e^c=egh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10005, #235323, #235324, … |
| maximal normal subgroup of | C2×#10005, #235323, #235324, … |
| maximal quotient of | #102753, #102757, #102758, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2993);