Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | Aut(G) order 393216 |
| homology | Abelianization (H1) = C24×C8 · Schur multiplier (H2) = C29 |
| direct product | C2×#2988 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C24 |
| commutator | C2 | C24×C8 |
| Frattini, cosocle | C2×C4 | C25 |
| socle | C23 | C23×C4 |
| prime radical | C22×#41 | C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1217 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)(11,12,13,15,14,16,17,18);(3,4)(5,8)(6,7)(9,10);(4,7)(8,10);(3,5,6,9)(4,8,7,10)136 conjugacy classes — view the full character table.
| Fitting subgroup of | C2×#2988⋊C3 |
| maximal subgroup of | #105600, #105602, #105646, … |
| maximal normal subgroup of | #105600, #105602, #105646, … |
| maximal quotient of | C2×#20543, C2×#20555, C2×#20569, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53243);