Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9980 |
| automorphism | Aut(G) = C2×#72278229 (order 4096) · Inn(G) = #392 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #392 |
| commutator, Frattini, cosocle | C23×C4 | C23 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
272 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 272 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)(3,6)(4,7)(5,8)(9,12)(10,13)(11,15)(14,16)(17,18,22,25)(19,26,28,31)(20,30,21,24)(23,29,32,27);(1,3,5,10)(2,6,8,13)(4,9,11,16)(7,14,15,12)(17,19,22,28)(18,23,25,32)(20,29,21,27)(24,31,30,26);(1,4)(2,7)(3,9)(5,11)(6,12)(8,15)(10,16)(13,14)(19,27)(20,21)(23,31)(24,30)(26,32)(28,29)Polycyclic presentation:
2^8|a^2=g,b^2=h,d^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,f^a=fgh,c^b=cf,d^b=dg,e^b=eg,f^b=fg,d^c=dh,f^c=fg28 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9984, #232512, #232513, … |
| maximal normal subgroup of | C2×#9984, #232512, #232513, … |
| maximal quotient of | #102476, #102477, #102488, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2972);