Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9519 |
| automorphism | Aut(G) = C23×#45355 (order 2048) · Inn(G) = #386 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | C4×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
198 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 198 conjugacy classes of subgroups — too many to draw. Only its 59 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:
(2,6)(3,9)(4,12)(8,14)(10,11)(13,15)(17,18,21,25,22,19,23,27)(20,26,29,32,28,24,30,31);(1,2,4,6,5,8,12,14)(3,7,10,13,11,16,9,15)(17,19,21,27,22,18,23,25)(20,26,30,31,28,24,29,32);(1,3,5,11)(2,7,8,16)(4,10,12,9)(6,13,14,15)(17,20)(18,24)(19,26)(21,30)(22,28)(23,29)(25,32)(27,31)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=g,d^2=g,e^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egh,e^b=eh,d^c=dh,e^c=eg,f^c=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9528, #202064, #202065, … |
| maximal normal subgroup of | C2×#9528, #202064, #202065, … |
| maximal quotient of | #100220, #100224, #100228, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2516);