Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #3415 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#3415 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D8 |
| commutator, Frattini, cosocle | C8 | C25 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#41 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
1063 conjugacy classes of subgroups; only the 479 normal subgroups are shown.
This group has 1063 conjugacy classes of subgroups — too many to draw. Only its 479 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4)(5,10)(6,9)(7,20)(8,13)(11,17)(12,18)(14,24)(15,21)(16,30)(19,23)(22,31)(25,29)(26,28)(27,32)(33,34);(4,9)(6,17)(7,18)(10,21)(11,12)(13,23)(15,29)(16,28)(19,31)(20,22)(24,32)(25,26)(27,34)(30,33);(3,5)(4,10)(6,15)(7,16)(8,14)(9,21)(11,25)(12,26)(13,24)(17,29)(18,28)(19,27)(20,30)(22,33)(23,32)(31,34);(5,14)(10,24)(15,27)(16,28)(21,32)(25,33)(26,30)(29,34)64 conjugacy classes — view the full character table.
| Fitting subgroup of | C2×#3415⋊C3 |
| maximal subgroup of | #152169, #152171, #152174, … |
| maximal normal subgroup of | #152169, #152171, #152174, … |
| maximal quotient of | C2×#33381, C2×#33410, C2×#33454, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55620);