Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9784 |
| automorphism | Aut(G) = C22×#72232559 (order 4096) · Inn(G) = #390 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #390 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C22×D4 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
192 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 192 conjugacy classes of subgroups — too many to draw. Only its 61 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:
(1,2,5,9,6,10,4,7)(3,11,14,16,12,15,13,8)(17,18,20,24,21,23,28,29)(19,22,26,31,25,30,27,32);(2,7,10,9)(3,12)(8,11,16,15)(13,14)(17,19,20,27,21,25,28,26)(18,22,29,31,23,30,24,32);(1,3,4,13,6,12,5,14)(2,8,9,11,10,16,7,15)(19,25)(22,30)(26,27)(31,32)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=e,d^2=gh,e^2=gh,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=egh,c^b=cg,d^b=dh,f^b=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9790, #205814, #205815, … |
| maximal normal subgroup of | C2×#9790, #205814, #205815, … |
| maximal quotient of | #102069, #102070, #102075, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2778);