Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24805 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
324 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 324 conjugacy classes of subgroups — too many to draw. Only its 115 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:
(1,2,4,5)(3,7,8,6)(9,10,14,17)(11,18,20,16)(12,13,23,15)(19,24,21,22);(2,5)(6,7)(9,11,15,19,14,20,13,21)(10,16,12,22,17,18,23,24);(9,12,13,17,14,23,15,10)(11,16,21,22,20,18,19,24);(1,3,4,8)(2,6,5,7)(9,13,14,15)(10,12,17,23)(11,19,20,21)(16,24,18,22)Polycyclic presentation:
2^8|a^2=e,b^2=g,c^2=f,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#28672, #6464093, #6464096, … |
| maximal normal subgroup of | C2×#28672, #6464093, #6464096, … |
| maximal quotient of | #324825, #324826, #324829, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21660);