Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29730 |
| automorphism | Aut(G) order 32768 |
| 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 | |
344 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 344 conjugacy classes of subgroups — too many to draw. Only its 121 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:
(1,2,4,5)(3,6,7,8)(9,10)(11,14)(12,13)(15,16);(2,5)(6,8)(17,18,20,23,21,24,29,32)(19,25,26,30,27,31,28,22);(1,3)(2,6)(4,7)(5,8)(10,13)(14,16)(17,19,21,27)(18,22,24,30)(20,28,29,26)(23,31,32,25);(3,7)(6,8)(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^8|a^2=e,b^2=g,c^2=h,d^2=f,g^2=h,b^a=be,c^a=cf,c^b=cg,d^c=de,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #169829, #169830, #169849, … |
| maximal normal subgroup of | #169829, #169830, #169849, … |
| maximal quotient of | #380883, #380911, #381148, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23326);