Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | C23 | C4×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
307 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 307 conjugacy classes of subgroups — too many to draw. Only its 167 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:
(1,2,5,8)(3,9,11,16)(4,12,14,7)(6,13,15,10)(17,18,22,25,23,19,21,27)(20,24,29,31,30,26,28,32);(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)(17,19,23,18)(20,26,30,24)(21,25,22,27)(28,31,29,32);(17,20,21,28,23,30,22,29)(18,24,27,32,19,26,25,31);(18,19)(24,26)(25,27)(31,32)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=fg,e^2=g,f^2=h,b^a=be,d^a=dh,e^a=eg,d^b=dh,e^b=eg88 conjugacy classes — view the full character table.
| maximal subgroup of | #122445, #122449, #122457, … |
| maximal normal subgroup of | #122445, #122449, #122457, … |
| maximal quotient of | #182457, #183473, #183885, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10946);