Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11522 |
| automorphism | Aut(G) = #72313219 (order 4096) · Inn(G) = #410 · Out(G) = D42 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #410 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×#41 | C23 |
| socle | C2 | #1799 |
| prime radical | #532 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C8 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
210 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 210 conjugacy classes of subgroups — too many to draw. Only its 63 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)(4,16)(5,10,19,8)(6,11)(7,14)(9,18)(13,26)(15,25,28,32)(17,23,30,24)(20,31,22,29)(21,27);(1,3,4,14,6,13,18,27)(2,7,9,12,11,21,16,26)(5,15,17,29,19,28,30,31)(8,22,23,32,10,20,24,25);(2,8)(3,13)(5,19)(7,20)(9,23)(10,11)(12,25)(14,27)(16,24)(17,30)(21,22)(26,32)Polycyclic presentation:
2^8|a^2=f,b^2=d,d^2=h,e^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eg,g^a=gh,c^b=ch,e^c=eh,f^c=fgh,f^e=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #130294, #130295, #130780, … |
| maximal normal subgroup of | #130294, #130295, #130780, … |
| maximal quotient of | #134237, #134261, #134291, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5796);