Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#2989 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C24 |
| commutator | C22 | C24×C4 |
| Frattini, cosocle | C2×C4 | C25 |
| socle | C23 | C23×C4 |
| prime radical | C22×#41 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1077 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 18 points:
(1,2);(4,7)(5,9)(11,12,13,16,14,15,17,18);(3,4)(5,8)(6,7)(9,10);(4,7)(8,10)(12,15)(16,18);(3,5,6,9)(4,8,7,10)88 conjugacy classes — view the full character table.
| maximal subgroup of | #105601, #105603, #105622, … |
| maximal normal subgroup of | #105601, #105603, #105622, … |
| maximal quotient of | C2×#20547, C2×#20549, C2×#20556, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53244);