Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #379 |
| prime radical | #537 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
355 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 355 conjugacy classes of subgroups — too many to draw. Only its 81 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 — 3 generators on 32 points:
(2,6)(4,11)(7,15)(8,12)(9,13)(14,16)(17,18,20,23,21,24,27,30)(19,22,26,29,25,28,31,32);(1,2)(3,7)(4,12)(5,8)(6,11)(9,14)(10,16)(13,15)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32);(1,3,5,10)(2,7,8,16)(4,13,11,9)(6,14,12,15)(19,25)(22,28)(26,31)(29,32)Polycyclic presentation:
2^8|a^2=f,c^2=g,d^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,c^b=ch,d^b=dg,e^b=eg,d^c=dg52 conjugacy classes — view the full character table.
| maximal subgroup of | #93706, #137273, #137274, … |
| maximal normal subgroup of | #93706, #137273, #137274, … |
| maximal quotient of | #96945, #96946, #96948, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1309);