Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23995 |
| automorphism | Aut(G) = C22×#511819 (order 2048) · Inn(G) = #535 · Out(G) = #120 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #535 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | #2598 |
| prime radical | #2017 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
323 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 323 conjugacy classes of subgroups — too many to draw. Only its 101 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 — 4 generators on 32 points:
(1,2)(3,13)(4,17)(5,10)(6,11)(7,12)(8,15)(9,19)(14,22)(16,23)(18,27)(20,24)(21,26)(25,29)(28,31)(30,32);(1,3,5,15,7,16,6,20)(2,8,11,13,12,24,10,23)(4,14,19,30,18,29,21,31)(9,25,17,32,26,22,27,28);(1,4)(2,9)(3,14)(5,19)(6,21)(7,18)(8,22)(10,27)(11,17)(12,26)(13,28)(15,30)(16,29)(20,31)(23,32)(24,25);(2,10,12,11)(4,18)(8,23,24,13)(9,17,26,27)(14,29)(19,21)(22,28,25,32)(30,31)Polycyclic presentation:
2^8|b^2=e,d^2=g,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,g^a=gh,c^b=cg,f^b=fh,d^c=dh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #485590, #485591, C2×#29387, … |
| maximal normal subgroup of | #485590, #485591, C2×#29387, … |
| maximal quotient of | #326615, #326616, #326617, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22375);