Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1913 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #386 |
| commutator | C23 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C2 | C22⋊#95 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
596 conjugacy classes of subgroups; only the 173 normal subgroups are shown.
This group has 596 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:
(1,2,5,10,6,9,17,23)(3,7,13,20,12,19,16,24)(4,8,15,21,14,22,28,32)(11,25,27,31,26,18,29,30);(1,3)(2,7,9,19)(4,11)(5,16)(6,12)(8,18,22,25)(10,24,23,20)(13,17)(14,26)(15,29)(21,31,32,30)(27,28);(1,4)(2,8)(3,11)(5,15)(6,14)(7,18)(9,22)(10,21)(12,26)(13,27)(16,29)(17,28)(19,25)(20,30)(23,32)(24,31)Polycyclic presentation:
2^8|a^2=f,b^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=gh,d^b=dh,e^b=eh,f^b=fh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | #146595, #146610, #146624, … |
| maximal normal subgroup of | #146595, #146610, #146624, … |
| maximal quotient of | #105753, #105754, #105757, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3744);