Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) order 524288 |
| homology | Abelianization (H1) = C22×C4×C8 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×C4×C4⋊C8 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C42 | C22 |
| commutator | C2 | C22×C4×C8 |
| Frattini, cosocle | C22×C4 | C24 |
| socle, prime radical | C24 | C22×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
775 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(3,4,5,6);(8,10)(11,12,13,15,14,16,17,18);(7,8,9,10)160 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7015, #94880, #94885, … |
| maximal normal subgroup of | C2×#7015, #94880, #94885, … |
| maximal quotient of | C2×C8×C4⋊C8, C2×#8262, C2×#8260, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13411);