Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19745 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#127 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #3184 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
364 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 364 conjugacy classes of subgroups — too many to draw. Only its 115 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:
(1,2,5,7)(3,9,11,16)(4,13,10,8)(6,14,15,12)(17,18)(19,25)(20,22)(21,29)(23,27)(24,28)(26,31)(30,32);(1,3,10,14,5,11,4,12)(2,6,13,16,7,15,8,9)(18,22)(23,30)(24,29)(25,31);(2,7)(6,15)(8,13)(9,16)(17,19,21,27,20,26,28,32)(18,23,24,25,22,30,29,31);(3,11)(6,15)(9,16)(12,14)(19,26)(23,30)(25,31)(27,32)Polycyclic presentation:
2^8|a^2=g,b^2=egh,c^2=fg,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,d^b=dg,d^c=dh40 conjugacy classes — view the full character table.
| maximal subgroup of | #143163, #232164, C2×#19754, … |
| maximal normal subgroup of | #143163, #232164, C2×#19754, … |
| maximal quotient of | #191534, #206317, #206318, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12742);