Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | Aut(G) = C25×D42 (order 2048) · Inn(G) = #120 · Out(G) = C26 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #385 |
| prime radical | C23 | #123 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
178 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 178 conjugacy classes of subgroups — too many to draw. Only its 63 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:
(2,6)(4,11)(7,15)(8,12)(9,13)(14,16)(17,18,21,25,22,20,29,24)(19,23,27,31,28,26,32,30);(1,2)(3,7)(4,12)(5,8)(6,11)(9,14)(10,16)(13,15)(17,19,21,27,22,28,29,32)(18,23,25,31,20,26,24,30);(1,3,5,10)(2,7,8,16)(4,13,11,9)(6,14,12,15)(17,20)(18,22)(19,26)(21,24)(23,28)(25,29)(27,30)(31,32)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=g,d^2=g,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,d^b=dg,e^b=eg,d^c=dg,e^c=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#8936, #193387, #193389, … |
| maximal normal subgroup of | C2×#8936, #193387, #193389, … |
| maximal quotient of | #99339, #99340, #99342, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1924);