Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#1442 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C25 | C23 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1471 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);(7,8,10,9)(12,13)(15,16,17,18);(4,6)(8,9)(11,12,14,13);(3,4)(5,6)(7,9,10,8)(11,13,14,12)88 conjugacy classes — view the full character table.
| maximal subgroup of | #109917, #109918, #116317, … |
| maximal normal subgroup of | #109917, #109918, #116317, … |
| maximal quotient of | C2×#7556, C2×#7557, C4×#1442, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6828);