Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32712 |
| automorphism | Aut(G) order 98304 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
424 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 424 conjugacy classes of subgroups — too many to draw. Only its 103 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)(3,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,20)(19,24,28,29)(21,30,22,25)(26,31,32,27);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,23,28)(18,24,20,29)(21,27,22,31)(25,32,30,26);(1,3)(2,6)(4,8)(5,7)(10,14)(13,16)(17,20,23,18)(19,26,28,32)(21,30,22,25)(24,31,29,27);(3,8)(5,6)(11,15)(13,16)(17,21,23,22)(18,25,20,30)(19,27,28,31)(24,32,29,26)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=h,d^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,g^b=gh,d^c=de,g^c=gh34 conjugacy classes — view the full character table.
| Fitting subgroup of | #32717⋊C3 |
| maximal subgroup of | #123034, #123044, C2×#32717, … |
| maximal normal subgroup of | #123034, #123044, C2×#32717, … |
| maximal quotient of | #391199, #391245, #391348, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25705);