Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9345 |
| automorphism | Aut(G) = C22×#72231762 (order 4096) · Inn(G) = #386 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
250 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 250 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)(3,11,13,8,14,15,12,16)(17,18)(19,22,26,29)(20,23)(21,24)(25,28,31,32)(27,30);(2,7,10,9)(8,11,16,15)(17,19)(18,22,24,29)(20,25)(21,26)(23,28,30,32)(27,31);(1,3,6,14)(2,8,10,16)(4,12,5,13)(7,11,9,15)(17,20,21,27)(18,23,24,30)(19,25,26,31)(22,28,29,32)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=g,d^2=gh,e^2=gh,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=egh,d^b=dh,f^b=fh,e^c=egh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9362, #200022, #200033, … |
| maximal normal subgroup of | C2×#9362, #200022, #200033, … |
| maximal quotient of | #100012, #100016, #100028, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2350);