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×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
269 conjugacy classes of subgroups; only the 89 normal subgroups are shown.
This group has 269 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:
(1,2,7,12,6,11,19,27)(3,4,9,16,14,8,20,23)(5,10,18,26,17,25,21,28)(13,22,29,32,30,15,24,31);(1,3)(2,8)(4,11)(5,13)(6,14)(7,20)(9,19)(10,15)(12,16)(17,30)(18,24)(21,29)(22,25)(23,27)(26,32)(28,31);(1,4)(2,9,11,20)(3,12,14,27)(5,15)(6,8)(7,16)(10,24,25,29)(13,28,30,26)(17,22)(18,31)(19,23)(21,32);(1,5,6,17)(2,10,11,25)(3,13,14,30)(4,15,8,22)(7,21,19,18)(9,24,20,29)(12,28,27,26)(16,32,23,31)Polycyclic presentation:
2^8|a^2=g,c^2=efh,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 | #6803849, #6803850, #6803853, … |
| maximal normal subgroup of | #6803849, #6803850, #6803853, … |
| maximal quotient of | #384265, #384296, #384429, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25687);