Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2199 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#2201 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×D4 | C24 |
| prime radical | C2×H(5,𝔽2) | C22 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ G | |
| LECS | G ▷ C2×D4 ▷ 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) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4)(5,13)(6,10)(7,11)(8,12)(9,15)(14,28)(16,29)(17,33)(18,25)(19,26)(20,34)(21,27)(22,30)(23,32)(24,31);(3,5,8,16)(4,9,12,23)(6,17,19,14)(7,15,21,32)(10,24,26,22)(11,13,27,29)(18,31,20,30)(25,33,34,28);(3,6,8,19)(4,10,12,26)(5,14,16,17)(7,20,21,18)(9,22,23,24)(11,25,27,34)(13,28,29,33)(15,31,32,30)34 conjugacy classes — view the full character table.
| maximal subgroup of | #152561, #153130, #153160, … |
| maximal normal subgroup of | #152561, #153130, #153160, … |
| maximal quotient of | C2×#12095, C2×#12101, C2×#12207, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26537);