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 | |
584 conjugacy classes of subgroups; only the 173 normal subgroups are shown.
This group has 584 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:
(1,2,5,11,7,10,6,19)(3,8,14,22,13,21,18,23)(4,9,16,24,15,25,17,30)(12,26,28,32,27,20,29,31);(1,3,6,14,7,13,5,18)(2,8,11,23,10,21,19,22)(4,12,17,28,15,27,16,29)(9,20,24,32,25,26,30,31);(1,4)(2,9)(3,12)(5,16)(6,17)(7,15)(8,20)(10,25)(11,24)(13,27)(14,28)(18,29)(19,30)(21,26)(22,31)(23,32)Polycyclic presentation:
2^8|a^2=f,b^2=g,f^2=h,g^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 | #146598, #146613, #146621, … |
| maximal normal subgroup of | #146598, #146613, #146621, … |
| maximal quotient of | #105755, #105759, #105760, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3745);