Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32706 |
| automorphism | Aut(G) order 49152 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
327 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 327 conjugacy classes of subgroups — too many to draw. Only its 113 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:
(1,2,4,7)(3,6,8,5)(9,10)(11,14)(12,13)(15,16);(1,3)(2,5)(4,8)(6,7)(10,13)(14,16)(17,18,20,23)(19,21,24,22);(1,3)(2,6)(4,8)(5,7)(10,13)(11,15)(18,21)(19,24)(22,23);(3,8)(5,6)(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^8|a^2=ef,b^2=h,d^2=f,g^2=h,b^a=be,c^a=cf,c^b=cg,d^b=def,g^b=gh,d^c=de,g^c=gh40 conjugacy classes — view the full character table.
| Fitting subgroup of | #32860⋊C3 |
| maximal subgroup of | C2×#32860, #6631557, #6631565, … |
| maximal normal subgroup of | C2×#32860, #6631557, #6631565, … |
| maximal quotient of | #391444, #391468, #391470, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25848);