Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C23 | C22×D4 |
| prime radical | C2×#532 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
825 conjugacy classes of subgroups; only the 255 normal subgroups are shown.
This group has 825 conjugacy classes of subgroups — too many to draw. Only its 255 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(6,8)(9,10,13,17,14,18,12,16)(11,19,20,15,22,23,21,24);(1,2,3,4)(9,11)(10,15)(12,20)(13,21)(14,22)(16,23)(17,19)(18,24);(5,6,7,8);(9,12,14,13)(10,16,18,17)(11,20,22,21)(15,23,24,19)Polycyclic presentation:
2^8|a^2=g,b^2=eg,c^2=f,d^2=h,e^2=h,g^2=h,b^a=be,c^a=cf,e^b=eh,g^b=gh76 conjugacy classes — view the full character table.
| maximal subgroup of | #107430, #107524, #110173, … |
| maximal normal subgroup of | #107430, #107524, #110173, … |
| maximal quotient of | #204485, #204495, #204509, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14245);