Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3068 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#121 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | C2×#390 |
| prime radical | C2×#575 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
730 conjugacy classes of subgroups; only the 237 normal subgroups are shown.
This group has 730 conjugacy classes of subgroups — too many to draw. Only its 237 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)(3,7,12,21)(4,14)(5,17)(6,9)(8,16)(10,18)(11,23,24,32)(13,27,26,20)(15,28)(19,30,31,25)(22,29);(1,3)(2,7,9,21)(4,11)(5,13)(6,12)(8,19,22,31)(10,20,17,27)(14,23,28,32)(15,24)(16,25)(18,26)(29,30);(1,4,5,16,6,15,18,29)(2,8,10,14,9,22,17,28)(3,11,13,25,12,24,26,30)(7,19,20,23,21,31,27,32);(4,15)(8,22)(11,24)(14,28)(16,29)(19,31)(23,32)(25,30)Polycyclic presentation:
2^8|a^2=g,b^2=eg,c^2=f,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,e^b=eh,g^b=gh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | #366636, #366641, #366642, … |
| maximal normal subgroup of | #366636, #366641, #366642, … |
| maximal quotient of | #204430, #204431, #204436, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14232);