Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | Aut(G) = C25×D42 (order 2048) · Inn(G) = #120 · Out(G) = C26 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #386 |
| prime radical | C23 | #121 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
216 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 71 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:
(2,6,8,11)(3,9,10,12)(17,18)(19,27,30,31)(20,24,23,29)(21,25)(22,26)(28,32);(1,2,4,11,5,8,13,6)(3,7,12,15,10,16,9,14)(17,19,28,29,22,30,21,24)(18,23,32,27,26,20,25,31);(1,3)(2,7)(4,12)(5,10)(6,14)(8,16)(9,13)(11,15)(17,20,22,23)(18,24,26,29)(19,25,30,32)(21,27,28,31)Polycyclic presentation:
2^8|a^2=f,b^2=dfh,c^2=h,d^2=g,e^2=gh,b^a=bd,c^a=ce,e^a=eh,d^b=dg,e^b=eg,f^b=fg,d^c=dg,e^c=egh,f^c=fg52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9056, #195474, #195478, … |
| maximal normal subgroup of | C2×#9056, #195474, #195478, … |
| maximal quotient of | #99596, #99598, #99603, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2044);