Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13096 |
| automorphism | Aut(G) = C2×#72236702 (order 2048) · Inn(G) = #1898 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1898 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×#41 | C23 |
| prime radical | C2×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
325 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 325 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)(3,12,17,30)(4,18,13,9)(5,10)(6,11)(7,15)(8,19,26,21)(14,25)(16,29,20,23)(22,27)(24,32)(28,31);(1,3,13,15,6,17,4,14)(2,7,18,30,11,25,9,12)(5,20,21,31,22,16,19,32)(8,23,10,28,26,29,27,24);(1,4,6,13)(2,8,11,26)(3,14,17,15)(5,21,22,19)(7,23,25,29)(9,27,18,10)(12,24,30,28)(16,32,20,31)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=h,e^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,d^b=dg,e^b=eh,f^b=fg,g^b=gh,d^c=dh,e^c=eh,f^c=fh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13097, #494748, #494750, … |
| maximal normal subgroup of | C2×#13097, #494748, #494750, … |
| maximal quotient of | #137685, #137690, #137730, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6085);