Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×#120 |
| prime radical | #537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
537 conjugacy classes of subgroups; only the 129 normal subgroups are shown.
This group has 537 conjugacy classes of subgroups — too many to draw. Only its 129 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)(12,20)(13,22)(14,18)(16,21)(17,23)(19,24);(2,5)(6,8)(9,11,14,12)(10,15,18,20)(13,21,23,19)(16,17,24,22);(9,12,14,11)(10,16,18,24)(13,19,23,21)(15,22,20,17);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^8|b^2=h,c^2=h,d^2=e,f^2=h,b^a=be,c^a=cf,f^a=fh,g^a=gh,c^b=cg,f^b=fh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #137074, #137076, #137085, … |
| maximal normal subgroup of | #137074, #137076, #137085, … |
| maximal quotient of | #241412, #241417, #241418, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16578);