Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2010 |
| automorphism | Aut(G) order 49152 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#2091 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #127 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | #399 |
| prime radical | C25 | Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
551 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 551 conjugacy classes of subgroups — too many to draw. Only its 167 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(3,4,6,10,7,9,12,16)(5,11,13,17,14,18,15,8);(3,5,12,13,7,14,6,15)(4,8,16,11,9,17,10,18);(4,9)(10,16)52 conjugacy classes — view the full character table.
| Fitting subgroup of | C2×#2091⋊C3, #72227816 |
| maximal subgroup of | #137211, #137212, #137213, … |
| maximal normal subgroup of | #137211, #137212, #137213, … |
| maximal quotient of | C2×#10659, C2×#10660, C2×#10661, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16710);