Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10496 |
| automorphism | Aut(G) = C23×D4×#120 (order 2048) · Inn(G) = #394 · Out(G) = C25 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #394 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | #123 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
163 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 163 conjugacy classes of subgroups — too many to draw. Only its 45 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) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,10)(3,4,8,14)(5,11,16,9)(7,15,13,12)(17,18)(19,22)(20,23)(21,25)(24,29)(26,30)(27,32)(28,31);(1,3)(2,7)(4,11)(5,15)(6,8)(9,14)(10,13)(12,16)(17,19,21,27)(18,22,25,32)(20,28,29,26)(23,31,24,30);(1,4,5,13,6,14,16,7)(2,8,9,15,10,3,11,12)(18,23)(20,29)(22,30)(24,25)(26,28)(31,32)Polycyclic presentation:
2^8|a^2=g,b^2=h,c^2=d,d^2=g,e^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgh,c^b=cf,d^b=dg,e^b=egh,e^c=egh,f^c=fg31 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10499, #240672, #240673, … |
| maximal normal subgroup of | C2×#10499, #240672, #240673, … |
| maximal quotient of | #104925, #104926, #104929, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3487);