Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1339 |
| automorphism | Aut(G) = #72282266 (order 2048) · Inn(G) = #327 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #327 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C2×M16 | C23 |
| socle | C2 | C2×#327 |
| prime radical | #3054 | C2 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
224 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 224 conjugacy classes of subgroups — too many to draw. Only its 61 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)(3,12,15,29)(4,9)(5,18,20,32)(6,22)(7,11)(8,14)(10,17,27,21)(13,28,16,26)(19,23)(24,30)(25,31);(1,3)(2,8)(4,13)(5,19)(6,16)(7,15)(9,23)(10,26)(11,25)(12,18)(14,21)(17,31)(20,30)(22,24)(27,28)(29,32);(1,4,7,6)(2,9,11,22)(3,13,15,16)(5,17,20,21)(8,23,25,24)(10,18,27,32)(12,28,29,26)(14,19,31,30)Polycyclic presentation:
2^8|a^2=e,c^2=h,d^2=gh,f^2=h,g^2=h,b^a=bd,d^a=df,g^a=gh,d^b=dg,e^b=efgh,f^b=fh,g^b=gh,e^d=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | #128773, #128806, #128834, … |
| maximal normal subgroup of | #128773, #128806, #128834, … |
| maximal quotient of | #122540, #122541, #122542, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4673);