Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1795 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #379 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C2 | C22⋊#95 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
574 conjugacy classes of subgroups; only the 173 normal subgroups are shown.
This group has 574 conjugacy classes of subgroups — too many to draw. Only its 173 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 — 3 generators on 32 points:
(1,2,5,12,7,11,6,10)(3,13,16,23,17,24,15,8)(4,18,20,26,21,9,19,25)(14,28,29,32,30,27,31,22);(1,3,6,15,7,17,5,16)(2,8,10,24,11,23,12,13)(4,14,20,29,21,30,19,31)(9,22,25,28,18,32,26,27);(1,4)(2,9)(3,14)(5,19)(6,20)(7,21)(8,22)(10,25)(11,18)(12,26)(13,27)(15,29)(16,31)(17,30)(23,32)(24,28)Polycyclic presentation:
2^8|a^2=f,b^2=g,d^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,d^b=dh,f^c=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | #146674, #146686, #146697, … |
| maximal normal subgroup of | #146674, #146686, #146697, … |
| maximal quotient of | #105598, #105599, #105600, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3755);