Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18810 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #3518 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
437 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 437 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:
(1,2,5,8)(3,9,10,16)(4,12,14,7)(6,13,15,11)(18,23)(19,27)(20,21)(25,26)(28,29)(30,31);(1,3,5,10)(2,6,8,15)(4,13,14,11)(7,16,12,9)(17,18,22,26)(19,24,28,32)(20,23,21,25)(27,30,29,31);(17,19,22,28)(18,24,26,32)(20,27,21,29)(23,30,25,31);(3,10)(6,15)(9,16)(11,13)(17,20,22,21)(18,25,26,23)(19,27,28,29)(24,31,32,30)Polycyclic presentation:
2^8|a^2=gh,b^2=g,c^2=h,d^2=h,e^2=g,f^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,d^b=dg,e^b=eg,f^b=fh,e^c=eh,f^c=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | #273703, #273720, #273731, … |
| maximal normal subgroup of | #273703, #273720, #273731, … |
| maximal quotient of | #168960, #169008, #169070, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11849);