Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31282 |
| automorphism | Aut(G) = #72338516 (order 4096) · Inn(G) = #2684 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2684 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#454 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
373 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 373 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:
(2,7)(3,11)(4,16)(5,19)(9,17)(10,23)(13,27)(14,15)(21,25)(22,26)(29,32)(30,31);(1,2,5,7,6,10,19,23)(3,12,11,21,15,8,14,25)(4,17,18,26,16,22,20,9)(13,29,30,28,31,32,27,24);(1,3)(2,8)(4,13)(5,14)(6,15)(7,21)(9,24)(10,12)(11,19)(16,31)(17,32)(18,27)(20,30)(22,29)(23,25)(26,28);(1,4,6,16)(2,9,10,26)(3,13,15,31)(5,20,19,18)(7,22,23,17)(8,24,12,28)(11,27,14,30)(21,29,25,32)Polycyclic presentation:
2^8|b^2=f,d^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fh,c^b=cg,d^b=de,g^b=gh,f^c=fh,e^d=eh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #153203, #153204, #153206, … |
| maximal normal subgroup of | #153203, #153204, #153206, … |
| maximal quotient of | #345809, #345811, #345813, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24271);