Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10018 |
| automorphism | Aut(G) = C2×#72278229 (order 4096) · Inn(G) = #392 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
331 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 331 conjugacy classes of subgroups — too many to draw. Only its 49 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)(3,7)(6,8)(9,10)(11,19)(12,15)(13,17)(14,18)(16,20)(21,23)(22,24);(1,2)(3,8)(4,6)(5,7)(10,15)(12,13)(16,19)(17,18)(20,22)(23,24);(3,7)(5,8)(9,11)(10,16)(12,20)(13,22)(14,21)(15,23)(17,19)(18,24)Polycyclic presentation:
2^8|d^2=g,e^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fgh,c^b=cf,d^b=dg,e^b=eg,d^c=dh,e^c=egh31 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10018, #235431, #235432, … |
| maximal normal subgroup of | C2×#10018, #235431, #235432, … |
| maximal quotient of | #103005, #103006, #103007, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3006);