Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12727 |
| automorphism | Aut(G) = #72311574 (order 4096) · Inn(G) = #1782 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1782 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C22×Q8 | C23 |
| prime radical | C2×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ #519 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
333 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 333 conjugacy classes of subgroups — too many to draw. Only its 69 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:
(1,2,6,11,7,12,3,13)(4,16,19,27,18,9,14,25)(5,10,15,23,21,26,20,8)(17,30,28,22,32,24,31,29);(1,3,7,6)(2,8,12,23)(4,14,18,19)(5,20,21,15)(9,22,16,29)(10,11,26,13)(17,31,32,28)(24,25,30,27);(1,4,7,18)(2,9)(3,14,6,19)(5,17,21,32)(8,22)(10,24)(11,25)(12,16)(13,27)(15,28,20,31)(23,29)(26,30)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=g,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,d^b=dh,f^b=fgh,e^c=eh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #94452, #152087, #152088, … |
| maximal normal subgroup of | #94452, #152087, #152088, … |
| maximal quotient of | #129297, #129301, #129378, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5718);