Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #21782 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2308 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3499 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
615 conjugacy classes of subgroups; only the 181 normal subgroups are shown.
This group has 615 conjugacy classes of subgroups — too many to draw. Only its 181 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) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,12,15,7)(4,16,18,9)(5,11)(6,10)(8,14,24,19)(13,21)(17,30,32,25)(20,22)(23,27)(26,28)(29,31);(1,3)(2,7)(4,13)(5,19)(6,15)(8,22)(9,21)(10,12)(11,24)(14,20)(16,26)(17,31)(18,28)(23,30)(25,29)(27,32);(1,3)(2,8)(4,13)(5,19)(6,15)(7,11)(9,23)(10,24)(12,22)(14,20)(16,29)(17,31)(18,28)(21,25)(26,30)(27,32);(1,4,6,18)(2,9,10,16)(3,13,15,28)(5,17,20,32)(7,21,12,26)(8,23,24,29)(11,25,22,30)(14,27,19,31)Polycyclic presentation:
2^8|a^2=g,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=ce,f^b=fh,g^b=gh,f^c=fh,g^c=gh,g^d=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #151823, #151824, #151831, … |
| maximal normal subgroup of | #151823, #151824, #151831, … |
| maximal quotient of | #213771, #213772, #213775, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14771);