Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1510 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#1515 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×C22⋊C4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
511 conjugacy classes of subgroups; only the 191 normal subgroups are shown.
This group has 511 conjugacy classes of subgroups — too many to draw. Only its 191 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 — 4 generators on 34 points:
(1,2);(3,4)(5,11,13,8)(6,9)(7,10)(12,17,18,15)(14,16)(19,20)(21,27)(22,30)(23,25)(24,28)(26,32)(29,34)(31,33);(3,5,7,13)(4,8,10,11)(6,12,14,18)(9,15,16,17)(19,21,23,29)(20,24,25,33)(22,31,32,28)(26,34,30,27);(3,6,7,14)(4,9,10,16)(5,12,13,18)(8,15,11,17)(20,25)(24,33)(26,30)(27,34)52 conjugacy classes — view the full character table.
| maximal subgroup of | #107309, #107595, #107615, … |
| maximal normal subgroup of | #107309, #107595, #107615, … |
| maximal quotient of | C2×#7596, C2×#7604, C2×#7605, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10545);