Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24269 |
| 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 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
315 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 315 conjugacy classes of subgroups — too many to draw. Only its 123 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)(3,7)(4,5)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,26,20,25,22,27)(19,28,30,32,29,23,31,24);(2,5)(6,7)(9,11,12,15)(10,13,14,16)(17,19,21,30,20,29,22,31)(18,23,26,24,25,28,27,32);(10,14)(13,16)(17,19,20,29)(18,24,25,32)(21,31,22,30)(23,27,28,26);(1,3,4,8)(2,6,5,7)Polycyclic presentation:
2^8|a^2=gh,b^2=f,c^2=h,d^2=eh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=deh,c^b=cg,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #311164, #311172, #311180, … |
| maximal normal subgroup of | #311164, #311172, #311180, … |
| maximal quotient of | #360403, #362555, #366061, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 20726);