Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29610 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #2006 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
528 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 528 conjugacy classes of subgroups — too many to draw. Only its 119 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,17)(13,21)(14,18)(16,19)(20,24)(22,23);(2,5)(6,8)(9,11,14,20)(10,15,18,24)(12,19,13,22)(16,21,23,17);(3,7)(6,8)(9,11)(10,16)(12,19)(13,22)(14,20)(15,21)(17,24)(18,23);(1,3,4,7)(2,6,5,8)(9,12,14,13)(10,17,18,21)(11,19,20,22)(15,16,24,23)Polycyclic presentation:
2^8|b^2=h,d^2=eh,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 | #487325, #487328, #487337, … |
| maximal normal subgroup of | #487325, #487328, #487337, … |
| maximal quotient of | #328186, #328187, #328194, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22671);