Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9519 |
| automorphism | Aut(G) = C23×D4×#120 (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 | C23×C4 | C23 |
| prime radical | #525 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
236 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 236 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:
(2,6)(3,9)(4,12)(8,14)(10,11)(13,15)(17,18)(19,22)(20,27,28,23)(21,24)(25,29,31,32)(26,30);(1,2,4,6,5,8,12,14)(3,7,10,13,11,16,9,15)(17,19,21,26)(18,22,24,30)(20,25,28,31)(23,29,27,32);(1,3)(2,7)(4,10)(5,11)(6,13)(8,16)(9,12)(14,15)(17,20)(18,23)(19,25)(21,28)(22,29)(24,27)(26,31)(30,32)Polycyclic presentation:
2^8|a^2=f,b^2=dfh,d^2=g,e^2=g,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×#9521, #201913, #201914, … |
| maximal normal subgroup of | C2×#9521, #201913, #201914, … |
| maximal quotient of | #100223, #100224, #100231, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2509);