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 | |
590 conjugacy classes of subgroups; only the 173 normal subgroups are shown.
This group has 590 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)(3,12)(4,16,18,9)(5,20)(6,10)(7,11)(8,14)(13,27,29,26)(15,24)(17,25,19,31)(21,23)(22,28,30,32);(1,3,6,14,7,15,5,21)(2,8,10,24,11,23,20,12)(4,13,19,28,18,29,17,32)(9,22,25,27,16,30,31,26);(1,4)(2,9)(3,13)(5,17)(6,19)(7,18)(8,22)(10,25)(11,16)(12,26)(14,28)(15,29)(20,31)(21,32)(23,30)(24,27)Polycyclic presentation:
2^8|a^2=f,b^2=g,d^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 | #146673, #146685, #146697, … |
| maximal normal subgroup of | #146673, #146685, #146697, … |
| maximal quotient of | #105598, #105599, #105602, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3754);