Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11757 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#99 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C22⋊C4 | C23 |
| prime radical | #392 | C22 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
216 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 49 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) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,13)(4,18,21,7)(5,11)(6,12)(8,16)(9,19,24,32)(10,22)(14,28)(15,25,20,26)(17,27)(23,30,29,31);(2,7)(3,14)(4,19)(8,25)(9,11)(10,24)(12,18)(13,23)(15,20)(16,17)(21,32)(26,28)(27,29)(30,31);(1,3,6,17)(2,8,12,28)(4,20,21,15)(5,16,22,14)(7,23,18,29)(9,25,24,26)(10,27,11,13)(19,30,32,31)Polycyclic presentation:
2^8|a^2=e,c^2=h,d^2=g,b^a=bd,c^a=cg,d^a=df,f^a=fh,c^b=cfh,d^b=dg,e^b=efg,d^c=dh,e^d=eh37 conjugacy classes — view the full character table.
| maximal subgroup of | #132111, #132112, #132114, … |
| maximal normal subgroup of | #132111, #132112, #132114, … |
| maximal quotient of | #124344, #124358, #124385, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4832);