Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18446 |
| automorphism | Aut(G) = C2×#72286737 (order 4096) · Inn(G) = C2×#120 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3560 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
465 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 465 conjugacy classes of subgroups — too many to draw. Only its 111 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 — 4 generators on 32 points:
(1,2,6,11)(3,8,13,16)(4,10,5,9)(7,14,15,12)(20,27)(22,28)(23,30)(25,31);(1,3)(2,7)(4,12)(5,14)(6,13)(8,9)(10,16)(11,15)(17,18)(19,22)(20,23)(21,24)(25,29)(26,28)(27,30)(31,32);(1,3)(2,8)(4,14)(5,12)(6,13)(7,9)(10,15)(11,16)(17,19)(18,22)(20,25)(21,26)(23,29)(24,28)(27,31)(30,32);(1,4,6,5)(2,9,11,10)(3,12,13,14)(7,8,15,16)(17,20,21,27)(18,23,24,30)(19,25,26,31)(22,29,28,32)Polycyclic presentation:
2^8|a^2=g,d^2=gh,e^2=g,b^a=be,c^a=cf,d^a=dgh,e^a=eg,e^b=eg,f^b=fh,d^c=dg,e^c=eh34 conjugacy classes — view the full character table.
| maximal subgroup of | #347418, #347419, #347420, … |
| maximal normal subgroup of | #347418, #347419, #347420, … |
| maximal quotient of | #168934, #168950, #168989, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11451);