Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1284 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #95 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | C2×#95 |
| prime radical | C24 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
273 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 273 conjugacy classes of subgroups — too many to draw. Only its 123 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) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,8,6,10,5,9)(3,11,12,7,14,15,13,16)(17,18,21,26,22,25,20,24)(19,27,28,32,29,23,30,31);(1,3,5,13,6,14,4,12)(2,7,8,15,10,16,9,11)(17,19)(18,23)(20,28)(21,30)(22,29)(24,31)(25,27)(26,32);(1,4,6,5)(2,8,10,9)(3,12,14,13)(7,15,16,11)(17,20,22,21)(18,24,25,26)(19,28,29,30)(23,31,27,32)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=g,d^2=gh,e^2=g,f^2=gh,b^a=bd,d^a=dg,f^a=fgh,e^b=eh64 conjugacy classes — view the full character table.
| maximal subgroup of | #93041, #93061, #105313, … |
| maximal normal subgroup of | #93041, #93061, #105313, … |
| maximal quotient of | #95067, #95070, #95080, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 974);