Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | Aut(G) = #72256639 (order 2048) · Inn(G) = C2×#99 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#99 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C2×#41 | C23 |
| socle | C2 | C4×#99 |
| prime radical | #532 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C8 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
226 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 226 conjugacy classes of subgroups — too many to draw. Only its 79 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 — 3 generators on 32 points:
(1,2,6,4,7,10,21,17)(3,11,14,25,15,26,29,32)(5,18,19,30,20,9,24,16)(8,22,12,27,23,31,28,13);(1,3)(2,8)(4,12)(5,13,20,27)(6,14)(7,15)(9,11,18,26)(10,23)(16,25,30,32)(17,28)(19,22,24,31)(21,29);(1,4,6,10,7,17,21,2)(3,12,14,23,15,28,29,8)(5,16,19,18,20,30,24,9)(11,13,25,22,26,27,32,31)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=e,d^2=h,e^2=h,b^a=bd,c^a=ch,d^a=dh,f^a=fgh,d^b=dg,g^b=gh,f^d=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #151945, #151948, #151953, … |
| maximal normal subgroup of | #151945, #151948, #151953, … |
| maximal quotient of | #128315, #128341, #128373, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5622);