Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2202 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#2203 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×Q8 | C24 |
| prime radical | C2×#454 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ G | |
| LECS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
546 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 546 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) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4)(5,14)(6,18)(7,10)(8,13)(9,15)(11,21)(12,23)(16,31)(17,29)(19,27)(20,28)(22,32)(24,30)(25,33)(26,34);(3,5,8,17)(4,9,13,27)(6,19,21,15)(7,22,23,16)(10,25,12,26)(11,29,18,14)(20,33,24,34)(28,31,30,32);(4,10)(7,23)(9,25)(11,30)(12,13)(14,31)(16,22)(18,28)(20,24)(26,27)(29,32)(33,34)34 conjugacy classes — view the full character table.
| maximal subgroup of | #152552, #153130, #153164, … |
| maximal normal subgroup of | #152552, #153130, #153164, … |
| maximal quotient of | C2×#12097, C2×#12098, C2×#12209, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26551);