Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12933 |
| automorphism | Aut(G) = #72311826 (order 2048) · Inn(G) = #1853 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1853 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
197 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 197 conjugacy classes of subgroups — too many to draw. Only its 73 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:
(1,2,5,12,7,13,6,11)(3,14,17,26,18,29,16,9)(4,19,21,28,23,27,22,10)(8,20,24,15,25,31,30,32);(1,3)(2,8,13,25)(4,20)(5,16)(6,17)(7,18)(9,10,26,28)(11,24,12,30)(14,27,29,19)(15,22)(21,32)(23,31);(1,3)(2,9)(4,15)(5,17)(6,16)(7,18)(8,10)(11,29)(12,14)(13,26)(19,24)(20,22)(21,31)(23,32)(25,28)(27,30)Polycyclic presentation:
2^8|a^2=f,b^2=fg,d^2=g,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,d^b=dg,f^b=fh,g^b=gh,e^c=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12934, #494277, #495739, … |
| maximal normal subgroup of | C2×#12934, #494277, #495739, … |
| maximal quotient of | #135832, #135848, #136012, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5922);