Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9443 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | #2433 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
517 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 517 conjugacy classes of subgroups — too many to draw. Only its 65 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(3,5,7,8)(4,6)(9,10,14,16,13,11,12,15);(1,3,4,7)(2,5)(6,8)(9,11,13,10)(12,15,14,16);(12,14)(15,16)Polycyclic presentation:
2^8|a^2=f,b^2=dfh,d^2=gh,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=egh,d^b=dh,e^b=egh,f^b=fh,d^c=dgh,f^c=fgh37 conjugacy classes — view the full character table.
| maximal subgroup of | #133390, #133393, #133395, … |
| maximal normal subgroup of | #133390, #133393, #133395, … |
| maximal quotient of | #100131, #100135, #100147, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2451);