Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9159 |
| automorphism | Aut(G) = C23×D43 (order 4096) · Inn(G) = #386 · Out(G) = C26 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C2×#41 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
216 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 69 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:
(1,2,7,10)(3,6,9,5)(4,11)(8,12)(13,15)(14,16)(17,18,21,25,22,26,20,24)(19,27,28,23,30,31,29,32);(1,3,5,2,7,9,6,10)(4,11,13,8,14,16,12,15)(17,19)(18,23)(20,28)(21,29)(22,30)(24,31)(25,27)(26,32);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,16)(10,15)(17,20,22,21)(18,24,26,25)(19,28,30,29)(23,31,32,27)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,c^2=h,d^2=gh,e^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=eg,f^b=fh,d^c=dg,e^c=eg,f^c=fg43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9170, #197124, #197129, … |
| maximal normal subgroup of | C2×#9170, #197124, #197129, … |
| maximal quotient of | #99793, #99794, #99807, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2158);