Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11799 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#99 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C4⋊C4 | C23 |
| prime radical | C24 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
193 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 193 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)(17,18,20,24,21,23,27,30)(19,22,26,28,25,29,32,31);(1,3)(2,6)(4,12)(5,11)(7,14)(8,16)(9,15)(10,13)(17,19)(18,22,23,29)(20,26)(21,25)(24,31,30,28)(27,32);(2,7)(6,14)(8,15)(9,16)(18,23)(19,25)(24,30)(26,32)Polycyclic presentation:
2^8|a^2=e,b^2=f,d^2=gh,e^2=h,b^a=bd,c^a=cg,d^a=df,f^a=fh,c^b=ch,d^b=dg,e^b=efgh,e^d=eh28 conjugacy classes — view the full character table.
| maximal subgroup of | #132981, #132982, #132983, … |
| maximal normal subgroup of | #132981, #132982, #132983, … |
| maximal quotient of | #124531, #124552, #124611, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4805);