Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29602 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
304 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 304 conjugacy classes of subgroups — too many to draw. Only its 111 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) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,5)(4,6)(7,8)(10,11)(14,15)(17,18,20,22)(19,23,26,29)(21,24,27,30)(25,32,28,31);(1,3,4,7)(2,5,6,8)(9,10,13,11)(12,14,16,15)(18,22)(19,25)(21,27)(23,31)(26,28)(29,32);(2,6)(5,8)(9,11,13,10)(12,14,16,15)(17,19,20,26)(18,23,22,29)(21,28,27,25)(24,31,30,32);(9,12,13,16)(10,14,11,15)(18,22)(19,26)(24,30)(25,28)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=e,d^2=eh,g^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gh,d^c=de,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#29948, #6435646, #6435652, … |
| maximal normal subgroup of | C2×#29948, #6435646, #6435652, … |
| maximal quotient of | #380030, #380203, #380321, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22936);