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) = C2×C4 |
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 | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
191 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 191 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) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,9,11,16)(4,8)(5,7)(6,10,14,12)(13,15)(17,18)(19,24)(20,28)(21,25,30,31)(22,26)(23,27,32,29);(1,3)(2,6)(4,12)(5,11)(7,14)(8,16)(9,15)(10,13)(17,19,22,20)(18,23,26,32)(21,29,30,27)(24,25,28,31);(2,7)(6,14)(8,15)(9,16)(17,20,22,19)(18,24,26,28)(21,27,30,29)(23,31,32,25)Polycyclic presentation:
2^8|a^2=e,b^2=h,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 | #132112, #132113, #132114, … |
| maximal normal subgroup of | #132112, #132113, #132114, … |
| maximal quotient of | #124346, #124356, #124387, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4834);