Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29610 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
469 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 469 conjugacy classes of subgroups — too many to draw. Only its 121 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:
(2,3)(6,7)(9,10)(11,14)(12,15)(13,17)(16,21)(18,22)(19,24)(20,23);(1,2)(3,5)(4,6)(7,8)(9,11,13,19)(10,14,17,24)(12,20,21,18)(15,23,16,22);(1,3,5,2)(4,6,8,7)(10,15)(12,21)(14,22)(16,17)(18,20)(23,24);(1,4,5,8)(2,6,3,7)Polycyclic presentation:
2^8|b^2=h,c^2=e,d^2=e,f^2=h,b^a=be,c^a=cf,f^a=fh,g^a=gh,c^b=cg,f^b=fh,d^c=de,f^c=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | #142501, #142503, #142533, … |
| maximal normal subgroup of | #142501, #142503, #142533, … |
| maximal quotient of | #328789, #328796, #328797, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22692);