Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10212 |
| automorphism | Aut(G) = C22×#72231160 (order 4096) · Inn(G) = #392 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | #532 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
232 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 232 conjugacy classes of subgroups — too many to draw. Only its 49 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) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,10)(4,6)(5,15)(7,9)(8,11)(12,16)(13,14)(19,26)(20,21)(23,27)(24,30)(28,29)(31,32);(1,3,6,12)(2,8,4,14)(5,13,7,11)(9,16,15,10)(17,18)(19,27)(20,24)(21,30)(22,25)(23,29)(26,31)(28,32);(1,4,6,2)(3,10,12,16)(5,15,7,9)(8,11,14,13)(17,19)(18,23)(20,29)(21,26)(22,28)(24,32)(25,31)(27,30)Polycyclic presentation:
2^8|b^2=g,c^2=g,d^2=g,e^2=h,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fh,c^b=cf,d^b=dg,f^b=fgh,d^c=dh,e^c=eh,f^c=fgh28 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10215, #232856, #232859, … |
| maximal normal subgroup of | C2×#10215, #232856, #232859, … |
| maximal quotient of | #102508, #102509, #102516, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3203);