Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2204 |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#2205 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×Q8 | C24 |
| prime radical | C2×2−1+4 | C22 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ G | |
| LECS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
568 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 568 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(3,4)(5,13)(6,16)(7,11)(8,18)(9,14)(10,15)(12,17);(3,5,9,8)(4,10,14,16)(6,17,15,7)(11,13,12,18);(3,6,9,15)(4,11,14,12)(5,7,8,17)(10,18,16,13)40 conjugacy classes — view the full character table.
| Fitting subgroup of | C2×#2205⋊C3, #72228026 |
| maximal subgroup of | #152558, #153124, #153140, … |
| maximal normal subgroup of | #152558, #153124, #153140, … |
| maximal quotient of | C2×#12101, C2×#12213, C2×#12130, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26554);