Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) = C2×#72310675 (order 4096) · Inn(G) = D4 · Out(G) = C2×#34736 |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
| direct product | C2×#2170 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C16 | D4 |
| commutator | C4 | C23×C8 |
| Frattini, cosocle | C2×C8 | C24 |
| socle | C22 | C8×D4 |
| prime radical | C2×#135 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C16 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
331 conjugacy classes of subgroups; only the 199 normal subgroups are shown.
This group has 331 conjugacy classes of subgroups — too many to draw. Only its 199 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);(3,4,8,12,7,13,22,30,9,14,24,31,23,11,27,29)(5,15,17,20,18,26,32,21,19,28,33,34,16,10,25,6);(3,5)(4,10)(6,12)(7,18)(8,17)(9,19)(11,28)(13,15)(14,26)(16,23)(20,30)(21,31)(22,32)(24,33)(25,27)(29,34);(3,6,8,15,7,20,22,26,9,21,24,28,23,34,27,10)(4,5,12,17,13,18,30,32,14,19,31,33,11,16,29,25)112 conjugacy classes — view the full character table.
| maximal subgroup of | #128673, #128674, #128675, … |
| maximal normal subgroup of | #128673, #128674, #128675, … |
| maximal quotient of | C2×C16×D8, C2×#12039, #483489, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26427);