Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×D8 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
183 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 183 conjugacy classes of subgroups — too many to draw. Only its 83 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(2,4)(5,6)(7,11)(8,9)(10,12)(13,14,17,22,25,28,24,19,16,20,15,21,18,23,26,27);(1,2,3,4)(6,9)(10,11)(14,19)(15,24)(17,26)(18,25)(20,27)(21,23)(22,28);(5,7,8,12)(6,10,9,11)(14,20)(19,27)(21,22)(23,28)Polycyclic presentation:
2^8|a^2=f,b^2=defg,c^2=eh,d^2=gh,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^b=dg,f^b=fg,g^b=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #131704, #131705, #131708, … |
| maximal normal subgroup of | #131704, #131705, #131708, … |
| maximal quotient of | #141857, #141873, #142098, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6265);