Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×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 | |
207 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 207 conjugacy classes of subgroups — too many to draw. Only its 107 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:
(1,2,3,4)(5,6)(7,11)(8,9)(10,12)(14,20)(15,16)(17,18)(21,26)(22,23)(24,27)(25,28);(2,4)(6,9)(10,11)(13,14,21,27,18,24,16,20,19,25,15,22,17,23,26,28);(5,7,8,12)(6,10,9,11)(13,15,18,26,19,21,17,16)(14,22,24,28,25,27,23,20)Polycyclic presentation:
2^8|a^2=f,b^2=def,c^2=eh,d^2=gh,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh76 conjugacy classes — view the full character table.
| maximal subgroup of | #131908, #131909, #131910, … |
| maximal normal subgroup of | #131908, #131909, #131910, … |
| maximal quotient of | #141884, #141911, #142001, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6270);