Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8516 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #379 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C2×#121 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
360 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 71 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 — 3 generators on 32 points:
(1,2,5,10,6,9,4,8)(3,11,12,16,13,7,14,15)(17,18)(19,26,28,23)(20,24)(21,25)(22,27,31,29)(30,32);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(17,19)(18,22)(20,29)(21,28)(23,24)(25,31)(26,32)(27,30);(1,4,6,5)(2,8,9,10)(3,12,13,14)(7,15,11,16)(17,19)(18,23)(20,27)(21,28)(22,24)(25,26)(29,30)(31,32)Polycyclic presentation:
2^8|a^2=f,c^2=g,d^2=h,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,d^b=dh,f^b=fgh,f^c=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #133113, #133114, #133115, … |
| maximal normal subgroup of | #133113, #133114, #133115, … |
| maximal quotient of | #97018, #97020, #97023, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1511);