Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | Aut(G) order 98304 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27 |
| direct product | C22×#344 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22⋊C4 |
| commutator | C22 | C22×C42 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C23 | #95 |
| prime radical | C25 | C2×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1095 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2);(3,4);(5,6)(7,13,14,10)(8,12)(9,11)(15,19,20,17)(16,18);(5,7,8,15,9,14,16,20)(6,10,12,17,11,13,18,19)88 conjugacy classes — view the full character table.
| Fitting subgroup of | A4×#344 |
| maximal subgroup of | #123440, #123441, #123443, … |
| maximal normal subgroup of | #123440, #123441, #123443, … |
| maximal quotient of | C22×#1281, C2×#7918, C2×#7919, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13347);