Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | Aut(G) order 32768 |
| 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 | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
346 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 346 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,5)(4,7)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,22)(19,25,27,23)(20,24,29,30)(26,28,32,31);(1,3)(2,4)(5,8)(6,7)(9,11,12,15)(10,13,14,16)(17,19)(18,23)(20,28)(21,27)(22,25)(24,26)(29,31)(30,32);(1,4,6,5)(2,7,8,3)(10,14)(13,16)(17,20,21,29)(18,24,22,30)(19,26,27,32)(23,31,25,28);(1,5,6,4)(2,7,8,3)(10,14)(13,16)(19,27)(23,25)(26,32)(28,31)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=e,d^2=eh,g^2=h,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,g^b=gh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #123249, #123265, #6633081, … |
| maximal normal subgroup of | #123249, #123265, #6633081, … |
| maximal quotient of | #383331, #384078, #384156, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24553);