Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12721 |
| automorphism | Aut(G) = C2×#72238017 (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, prime radical, cosocle | C2×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
335 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 335 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)(3,12)(4,16,18,9)(5,10)(6,11)(7,13,23,30)(8,17,24,32)(14,27)(15,28)(19,21)(20,25)(22,29,26,31);(1,3,6,15)(2,7,11,23)(4,13,18,30)(5,19,20,14)(8,21,24,27)(9,12,16,28)(10,26,25,22)(17,31,32,29);(1,4,6,18)(2,8)(3,13,15,30)(5,17,20,32)(7,21)(9,25)(10,16)(11,24)(12,22)(14,29,19,31)(23,27)(26,28)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=gh,e^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,e^b=eh,f^b=fg,d^c=dh,f^c=fh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #94452, #152147, #152149, … |
| maximal normal subgroup of | #94452, #152147, #152149, … |
| maximal quotient of | #129292, #129299, #129374, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5712);