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) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
429 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 429 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:
(1,2,4,6)(3,5,7,8)(9,10,12,14)(11,13,15,16)(17,18,21,23)(19,26,28,24)(20,29,22,25)(27,31,30,32);(1,2)(3,5)(4,6)(7,8)(9,11,12,15)(10,13,14,16)(18,23)(19,27)(20,22)(24,31)(26,32)(28,30);(1,3,4,7)(2,5,6,8)(10,14)(13,16)(17,19)(18,24)(20,27)(21,28)(22,30)(23,26)(25,31)(29,32);(2,6)(5,8)(10,14)(13,16)(17,20,21,22)(18,25,23,29)(19,27,28,30)(24,31,26,32)Polycyclic presentation:
2^8|a^2=efh,b^2=fh,c^2=eh,d^2=h,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 | #123111, #123121, #6634628, … |
| maximal normal subgroup of | #123111, #123121, #6634628, … |
| maximal quotient of | #383684, #383951, #384197, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24725);