Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10260 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #121 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
281 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 281 conjugacy classes of subgroups — too many to draw. Only its 79 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(5,6,10,7)(8,11,9,12)(13,14,17,20)(15,21,23,28)(16,24,26,19)(18,25,27,22);(2,4)(6,11)(7,12)(8,9);(5,7,10,6)(8,12,9,11)(13,15,17,23)(14,18,20,27)(16,25,26,22)(19,28,24,21)Polycyclic presentation:
2^8|a^2=gh,c^2=gh,d^2=g,e^2=h,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fg,c^b=cf,d^b=dg,f^b=fg,d^c=dg,e^c=eh,f^c=fg46 conjugacy classes — view the full character table.
| maximal subgroup of | #93796, #93838, #125875, … |
| maximal normal subgroup of | #93796, #93838, #125875, … |
| maximal quotient of | #102461, #102468, #102571, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3266);