Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23387 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
607 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 607 conjugacy classes of subgroups — too many to draw. Only its 175 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:
(1,2,3,4)(9,10,14,18)(11,19,22,24)(12,17,13,16)(15,20,23,21);(5,6)(7,8)(9,11,13,21,14,22,12,20)(10,15,17,19,18,23,16,24);(2,4)(6,8)(10,16)(11,20)(12,13)(15,23)(17,18)(21,22);(9,12,14,13)(10,17,18,16)(11,20,22,21)(15,19,23,24)Polycyclic presentation:
2^8|a^2=ef,b^2=e,d^2=h,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gh,c^b=cg,d^c=dh,e^c=eh,f^c=fh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #395733, #395946, #395970, … |
| maximal normal subgroup of | #395733, #395946, #395970, … |
| maximal quotient of | #242421, #242425, #242430, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16600);