Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#1547 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×C22⋊C4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| prime radical | C2×#532 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
677 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(4,8)(5,6)(9,10)(11,12,14,16,15,18,13,17);(3,4)(5,8)(6,10)(7,9)(12,16)(13,14)(17,18);(3,5,7,6)(4,8,9,10)64 conjugacy classes — view the full character table.
| maximal subgroup of | #106218, #107370, #107634, … |
| maximal normal subgroup of | #106218, #107370, #107634, … |
| maximal quotient of | C2×#7574, C2×#7586, #160492, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10691);