Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1655 |
| automorphism | Aut(G) order 98304 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#1655 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #120 |
| commutator, Frattini, cosocle | C42 | C24 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
665 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 665 conjugacy classes of subgroups — too many to draw. Only its 119 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) = 64 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(4,8)(5,6)(9,10)(12,15)(13,17)(16,18);(3,4)(5,10)(6,8)(7,9)(11,12)(13,18)(14,16)(15,17);(3,4)(5,8)(6,10)(7,9)(13,17)(15,18)40 conjugacy classes — view the full character table.
| Fitting subgroup of | C2×#1655⋊C3 |
| maximal subgroup of | #152575, #153159, #153163, … |
| maximal normal subgroup of | #152575, #153159, #153163, … |
| maximal quotient of | C2×#7734, C2×#7742, C2×#7822, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11966);