Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18930 |
| 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 | #532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
429 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 429 conjugacy classes of subgroups — too many to draw. Only its 113 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:
(1,2,7,4)(3,10,13,16)(5,9,6,14)(8,15,11,12)(17,18)(19,25)(20,28)(21,23)(22,27)(24,29)(26,32)(30,31);(1,3)(2,8)(4,11)(5,12)(6,15)(7,13)(9,16)(10,14)(20,29)(24,28)(25,32)(27,30);(1,4,7,2)(3,11,13,8)(5,9,6,14)(10,12,16,15)(17,19)(18,22)(20,30)(21,26)(23,31)(24,32)(25,28)(27,29);(1,5,7,6)(2,9,4,14)(3,12,13,15)(8,16,11,10)(18,23)(22,31)(24,28)(25,32)Polycyclic presentation:
2^8|a^2=g,c^2=g,d^2=g,e^2=g,f^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | #245317, #245321, #245337, … |
| maximal normal subgroup of | #245317, #245321, #245337, … |
| maximal quotient of | #170390, #170437, #170498, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11927);