Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23391 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
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 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
407 conjugacy classes of subgroups; only the 159 normal subgroups are shown.
This group has 407 conjugacy classes of subgroups — too many to draw. Only its 159 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:
(1,2,5,7)(3,8,4,6)(9,10)(11,13)(12,14)(15,16)(17,18)(19,25)(20,22)(21,24)(23,27)(26,31)(28,30)(29,32);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(18,22)(19,26)(21,28)(23,31)(25,32)(27,29);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(17,19,20,27)(18,23,22,25)(21,29,28,26)(24,31,30,32);(19,27)(23,25)(26,29)(31,32)Polycyclic presentation:
2^8|a^2=eh,b^2=ef,c^2=e,g^2=h,b^a=be,c^a=cf,c^b=cg,g^b=gh,d^c=dh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #401697, #401698, #401711, … |
| maximal normal subgroup of | #401697, #401698, #401711, … |
| maximal quotient of | #244086, #244089, #244093, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16734);