Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#525 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
977 conjugacy classes of subgroups; only the 247 normal subgroups are shown.
This group has 977 conjugacy classes of subgroups — too many to draw. Only its 247 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) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6)(7,8)(9,10,13,18,14,17,12,16)(11,19,20,15,22,23,21,24);(1,2,3,4)(9,11)(10,15)(12,20)(13,21)(14,22)(16,23)(17,24)(18,19);(6,8)(9,12,14,13)(10,16,17,18)(11,20,22,21)(15,23,24,19);(10,17)(15,24)(16,18)(19,23)Polycyclic presentation:
2^8|a^2=g,b^2=eg,c^2=h,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,e^b=eh,g^b=gh64 conjugacy classes — view the full character table.
| maximal subgroup of | #107948, #119383, #187321, … |
| maximal normal subgroup of | #107948, #119383, #187321, … |
| maximal quotient of | #193424, #193435, #193443, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13898);