Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11753 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
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 | C25 | D4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
253 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 253 conjugacy classes of subgroups — too many to draw. Only its 51 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)(3,11,13,25)(4,15,17,9)(5,8)(6,10)(7,12,20,30)(14,27,31,32)(16,21,29,22)(18,26,28,23)(19,24);(1,3,6,14)(2,7,10,21)(4,16,17,30)(5,13,19,31)(8,22,24,20)(9,25,15,32)(11,26,27,23)(12,28,29,18);(2,8)(4,17)(7,20)(9,26)(10,24)(11,25)(12,29)(15,23)(16,30)(18,28)(21,22)(27,32)Polycyclic presentation:
2^8|a^2=e,b^2=h,d^2=g,b^a=bd,c^a=cg,d^a=df,f^a=fh,c^b=cf,d^b=dg,e^b=efg,d^c=dh,e^d=eh37 conjugacy classes — view the full character table.
| maximal subgroup of | #132088, #132089, #132090, … |
| maximal normal subgroup of | #132088, #132089, #132090, … |
| maximal quotient of | #124341, #124361, #124384, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4829);