Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19384 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
321 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 321 conjugacy classes of subgroups — too many to draw. Only its 121 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 — 4 generators on 32 points:
(1,2,5,8)(3,9,12,16)(4,13,10,7)(6,14,15,11)(17,18,21,25)(19,22,26,30)(20,23,28,24)(27,29,32,31);(1,3,10,14,5,12,4,11)(2,6,13,16,8,15,7,9)(17,19)(18,22)(20,27)(21,26)(23,29)(24,31)(25,30)(28,32);(18,23)(20,28)(22,29)(24,25)(27,32)(30,31);(19,26)(22,30)(27,32)(29,31)Polycyclic presentation:
2^8|a^2=gh,b^2=eg,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,d^b=dh,f^c=fh49 conjugacy classes — view the full character table.
| maximal subgroup of | #267868, #267874, #267892, … |
| maximal normal subgroup of | #267868, #267874, #267892, … |
| maximal quotient of | #189133, #197852, #197928, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12381);