Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#99 |
| commutator | C2×C4 | C2×C42 |
| Frattini, prime radical, cosocle | C22×D4 | C23 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
282 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 282 conjugacy classes of subgroups — too many to draw. Only its 81 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,4,3)(5,6)(7,13)(8,16)(9,11)(10,14)(12,17)(15,19)(18,20);(5,7)(6,10)(8,14,17,20)(9,15)(11,18)(12,13,16,19);(1,3,4,2)(6,11)(10,18)(12,16)(13,19)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=e,d^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 | #130483, #130485, #130522, … |
| maximal normal subgroup of | #130483, #130485, #130522, … |
| maximal quotient of | #128312, #128339, #128370, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5615);