Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9331 |
| automorphism | Aut(G) = C23×#35346 (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 | C2×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
267 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 267 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) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,12,18,30,3,15)(4,20,23,31,22,28,19,26)(6,13,16,8,7,14,17,10)(9,27,25,24,11,32,29,21);(2,8,13,12)(3,16)(4,21)(5,17)(9,25,20,31)(10,14,15,30)(11,29,28,26)(19,23)(22,24)(27,32);(2,9)(3,17)(4,22)(5,16)(8,25)(10,29)(11,30)(12,31)(13,20)(14,28)(15,26)(19,27)(21,24)(23,32)Polycyclic presentation:
2^8|a^2=f,b^2=df,d^2=gh,e^2=gh,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eh,d^b=dh,e^b=eg,f^b=fh,d^c=dg,e^c=egh,f^c=fg34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9341, #199718, #199722, … |
| maximal normal subgroup of | C2×#9341, #199718, #199722, … |
| maximal quotient of | #99995, #99997, #100003, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2329);