Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32696 |
| automorphism | Aut(G) = C2×#72236189 (order 2048) · Inn(G) = #2749 · Out(G) = C24 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2749 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
264 conjugacy classes of subgroups; only the 89 normal subgroups are shown.
This group has 264 conjugacy classes of subgroups — too many to draw. Only its 89 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:
(1,2,7,11,6,12,24,32)(3,8,16,28,15,27,25,30)(4,17,20,21,19,9,29,14)(5,10,23,13,22,18,26,31);(1,3)(2,8)(4,18)(5,21)(6,15)(7,25)(9,26)(10,19)(11,30)(12,27)(13,20)(14,22)(16,24)(17,23)(28,32)(29,31);(1,4,6,19)(2,9)(3,13)(5,8,22,27)(7,20,24,29)(10,25)(11,14)(12,17)(15,31)(16,18)(21,32)(23,28,26,30);(1,5,6,22)(2,10,12,18)(3,14,15,21)(4,8,19,27)(7,26,24,23)(9,25,17,16)(11,31,32,13)(20,30,29,28)Polycyclic presentation:
2^8|a^2=g,c^2=ef,d^2=h,g^2=h,b^a=be,c^a=cf,d^a=def,e^a=eh,c^b=cg,d^b=de,f^b=fh,g^b=gh,e^c=eh,f^c=fh,g^d=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | #6803848, #6803852, #6803853, … |
| maximal normal subgroup of | #6803848, #6803852, #6803853, … |
| maximal quotient of | #384265, #384294, #384428, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25685);