Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10012 |
| automorphism | Aut(G) = C22×#72210469 (order 4096) · Inn(G) = #127 · Out(G) = C22×#120 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #127 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #399 |
| prime radical | C23 | C22⋊Q8 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
152 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 152 conjugacy classes of subgroups — too many to draw. Only its 53 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) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,9,6,10,4,8)(3,11,12,16,13,15,14,7)(17,18,20,24,21,25,29,22)(19,26,27,31,28,32,30,23);(1,3,4,12,6,13,5,14)(2,7,8,11,10,16,9,15)(18,22)(20,29)(23,26)(24,25)(27,30)(31,32);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,15,16,11)(17,19,21,28)(18,23,25,31)(20,30,29,27)(22,26,24,32)Polycyclic presentation:
2^8|a^2=d,b^2=deg,c^2=g,d^2=g,e^2=h,b^a=bd,c^a=ce,f^a=fgh,c^b=cf,d^b=dg,e^b=eg,d^c=dh,e^c=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10664, #243830, #243832, … |
| maximal normal subgroup of | C2×#10664, #243830, #243832, … |
| maximal quotient of | #105233, #105234, #105237, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3652);