Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1347 |
| automorphism | Aut(G) = C2×C4×#3003 (order 1024) · Inn(G) = #102 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #102 |
| commutator | C2×C8 | C2×C8 |
| Frattini, cosocle | C2×C4×C8 | C22 |
| socle | C22 | D4⋊C8 |
| prime radical | C2×#135 | C4 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
129 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 129 conjugacy classes of subgroups — too many to draw. Only its 37 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 — 2 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10,13,18,15,19,24,17,16,20,12,23,14,11,21,22);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,17)(13,22)(14,20)(16,19)(18,24)(21,23)Polycyclic presentation:
2^8|a^2=d,c^2=fh,d^2=g,e^2=h,f^2=h,g^2=h,b^a=bc,c^a=ce,e^a=eh,f^a=fh,c^b=cf,d^b=defh,e^b=eh,f^b=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7114, #124074, #124103, … |
| maximal normal subgroup of | C2×#7114, #124074, #124103, … |
| maximal quotient of | #92821, #92822, #92837, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 102);