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 | #3500 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
639 conjugacy classes of subgroups; only the 181 normal subgroups are shown.
This group has 639 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)(3,11,14,7)(4,15,17,27)(5,10)(6,9)(8,16,24,19)(12,28)(13,26,18,22)(20,25)(21,29)(23,31)(30,32);(1,3)(2,7)(4,12)(5,18)(6,14)(8,23)(9,11)(10,26)(13,20)(15,32)(16,31)(17,30)(19,29)(21,24)(22,25)(27,28);(1,4,6,17)(2,8,9,24)(3,12,14,30)(5,19,20,16)(7,21,11,23)(10,27,25,15)(13,31,18,29)(22,28,26,32);(1,3)(2,7)(4,12)(5,13)(6,14)(8,21)(9,11)(10,22)(15,32)(16,29)(17,30)(18,20)(19,31)(23,24)(25,26)(27,28)Polycyclic presentation:
2^8|a^2=g,c^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 | #151825, #151829, #151830, … |
| maximal normal subgroup of | #151825, #151829, #151830, … |
| maximal quotient of | #213770, #213773, #213775, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14772);