Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30056 |
| automorphism | Aut(G) = C2×#72236189 (order 2048) · Inn(G) = #2631 · Out(G) = C24 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2631 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #2008 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
342 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 342 conjugacy classes of subgroups — too many to draw. Only its 97 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:
(1,2,5,10,6,11,19,8)(3,12,14,22,16,20,15,7)(4,9,17,24,18,25,30,23)(13,26,27,32,29,31,28,21);(1,3)(2,7)(4,13)(5,15)(6,16)(8,12)(9,21)(10,20)(11,22)(14,19)(17,28)(18,29)(23,26)(24,31)(25,32)(27,30);(2,8)(4,17)(5,19)(7,20)(10,11)(12,22)(13,27)(14,15)(18,30)(23,24)(26,31)(28,29);(1,4,5,17,6,18,19,30)(2,9,10,24,11,25,8,23)(3,13,15,28,16,29,14,27)(7,21,20,31,22,32,12,26)Polycyclic presentation:
2^8|a^2=e,d^2=f,e^2=h,f^2=h,b^a=be,c^a=cf,g^a=gh,c^b=cg,e^b=eh,d^c=de,e^c=eh,f^c=fh,g^d=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#30056, #6434129, #6434130, … |
| maximal normal subgroup of | C2×#30056, #6434129, #6434130, … |
| maximal quotient of | #331007, #331010, #331012, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23044);