Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29742 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
301 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 301 conjugacy classes of subgroups — too many to draw. Only its 115 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) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,5)(4,6)(7,8)(9,10,13,15,14,11,12,16)(17,18,21,22)(19,23,26,29)(20,24,28,30)(25,31,27,32);(1,3,4,7)(2,5,6,8)(9,11,13,16,14,10,12,15)(18,22)(23,29)(24,30)(31,32);(2,6)(5,8)(9,11)(10,14)(12,16)(13,15)(17,19,21,26)(18,23,22,29)(20,27,28,25)(24,32,30,31);(9,12,14,13)(10,15,11,16)(17,20,21,28)(18,24,22,30)(19,25,26,27)(23,31,29,32)Polycyclic presentation:
2^8|a^2=eg,b^2=f,c^2=e,d^2=eh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=dh,d^c=de,f^c=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #102691, #102697, #102698, … |
| maximal normal subgroup of | #102691, #102697, #102698, … |
| maximal quotient of | #380220, #380734, #381039, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23223);