Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | #3391 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×Q8 ◁ C8×Q8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
259 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 259 conjugacy classes of subgroups — too many to draw. Only its 101 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) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(6,7)(10,14)(11,17)(12,19)(15,22)(16,24)(18,21)(20,23);(1,2,4,5)(3,7,8,6)(10,15)(14,22)(16,21)(18,24);(9,10,13,15)(11,18,20,24)(12,21,19,16)(14,23,22,17);(1,3,4,8)(2,6,5,7)Polycyclic presentation:
2^8|b^2=e,c^2=h,d^2=e,f^2=gh,g^2=h,b^a=be,c^a=cf,f^a=fg,g^a=gh,c^b=ch,d^b=de,f^c=fg,g^c=gh43 conjugacy classes — view the full character table.
| maximal subgroup of | #491814, #491823, #491830, … |
| maximal normal subgroup of | #491814, #491823, #491830, … |
| maximal quotient of | #497614, #497669, #497957, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26923);