Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9263 |
| automorphism | Aut(G) = C23×#2403 (order 1024) · Inn(G) = #386 · Out(G) = C24 |
| 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 | #3188 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
252 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 252 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) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,7,10,16)(3,12,14,24)(4,8)(11,22,21,23)(13,31,17,15)(19,25)(20,30)(26,27);(1,2,8,23,18,21,19,7)(3,9,24,27,31,32,13,20)(4,16,6,10,25,22,5,11)(12,30,15,28,17,26,14,29);(1,3)(2,9)(4,17)(5,15)(6,14)(7,20)(8,24)(10,29)(11,28)(12,25)(13,19)(16,26)(18,31)(21,32)(22,30)(23,27)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,d^2=gh,e^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=eh,f^b=fh,d^c=dh,e^c=egh,f^c=fh43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9264, #198494, #198496, … |
| maximal normal subgroup of | C2×#9264, #198494, #198496, … |
| maximal quotient of | #99917, #99918, #99921, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2252);