Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | Aut(G) order 1572864 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
| direct product | C2×#2825 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C24 |
| commutator, Frattini, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1465 conjugacy classes of subgroups; only the 447 normal subgroups are shown.
This group has 1465 conjugacy classes of subgroups — too many to draw. Only its 447 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2);(4,7)(8,10)(11,12)(13,16)(14,15)(17,18)(19,20,23,24)(21,25,26,22);(5,9)(8,10)(12,15)(13,17)(21,26)(22,25);(3,4)(5,8)(6,7)(9,10)(11,12)(13,18)(14,15)(16,17)(19,21)(20,22)(23,26)(24,25);(3,5)(4,8)(6,9)(7,10)(11,13)(12,16)(14,17)(15,18)(19,22)(20,21)(23,25)(24,26)52 conjugacy classes — view the full character table.
| Fitting subgroup of | C2×#2825⋊C3 |
| maximal subgroup of | C4×#2825, #549633, #549636, … |
| maximal normal subgroup of | C4×#2825, #549633, #549636, … |
| maximal quotient of | C2×#14621, C2×#14637, C2×#14642, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29651);