Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#1934 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
731 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);(3,4)(5,6)(7,8,10,9)(12,13)(14,15)(17,18);(7,9,10,8)(11,12,15,17,16,18,14,13);(4,6)(11,13,14,18,16,17,15,12)88 conjugacy classes — view the full character table.
| maximal subgroup of | #102062, #102066, #102070, … |
| maximal normal subgroup of | #102062, #102066, #102070, … |
| maximal quotient of | C2×#9554, C2×#9558, #202567, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14070);