Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11521 |
| automorphism | Aut(G) = #72303760 (order 2048) · Inn(G) = #410 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #410 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×M16 | C23 |
| socle | C2 | #1464 |
| prime radical | C2×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C42 ◁ C4×M16 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
144 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 144 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:
(1,2,5,11,6,10,21,27,7,12,20,28,4,9,18,26)(3,13,16,29,15,8,22,24,14,23,19,31,17,25,30,32);(1,3)(2,8)(4,17)(5,19)(6,15)(7,14)(9,13)(10,23)(11,24)(12,25)(16,20)(18,22)(21,30)(26,29)(27,31)(28,32);(2,9,12,10)(3,14)(5,20)(8,23,25,13)(11,27,28,26)(15,17)(18,21)(24,29,32,31)Polycyclic presentation:
2^8|a^2=e,c^2=f,d^2=h,e^2=g,g^2=h,b^a=bd,c^a=cg,d^a=df,f^a=fh,c^b=ch,d^b=dh,e^b=efh,e^c=eh,e^d=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | #94192, #94193, #133267, … |
| maximal normal subgroup of | #94192, #94193, #133267, … |
| maximal quotient of | #124680, #124707, #124730, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4512);