Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| socle | C22 | C22×C4⋊C4 |
| prime radical | C2×#41 | Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
306 conjugacy classes of subgroups; only the 189 normal subgroups are shown.
This group has 306 conjugacy classes of subgroups — too many to draw. Only its 189 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,4)(3,10,14,16)(5,9,6,15)(8,12,11,13)(17,18,21,25,22,26,20,24)(19,27,29,32,30,31,28,23);(1,3,5,12,7,14,6,13)(2,8,15,16,4,11,9,10)(17,19,21,29,22,30,20,28)(18,23,24,31,26,32,25,27);(1,4,7,2)(3,11,14,8)(5,9,6,15)(10,13,16,12)(17,20,22,21)(18,24,26,25)(19,28,30,29)(23,31,32,27);(1,5,7,6)(2,9,4,15)(3,12,14,13)(8,10,11,16)(17,20,22,21)(18,24,26,25)(19,28,30,29)(23,31,32,27)Polycyclic presentation:
2^8|a^2=f,b^2=eg,c^2=g,d^2=g,e^2=g,f^2=h,b^a=be,e^a=eg,d^c=dgh70 conjugacy classes — view the full character table.
| Fitting subgroup of | #17837⋊C3 |
| maximal subgroup of | #136455, #136457, #137042, … |
| maximal normal subgroup of | #136455, #136457, #137042, … |
| maximal quotient of | #180938, #183421, #183611, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10825);