Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29742 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
386 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 386 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,24,21,25,20,26)(19,23,27,32,28,30,29,31);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19,20,29,21,28,22,27)(18,23,26,31,25,30,24,32);(1,3)(2,6)(4,7)(5,8)(10,14)(13,16)(18,24)(19,27)(20,22)(23,30)(25,26)(28,29);(3,7)(6,8)(18,25)(19,28)(24,26)(27,29)Polycyclic presentation:
2^8|a^2=gh,b^2=f,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=dh,d^c=de,f^c=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#30198, #6433735, #6433738, … |
| maximal normal subgroup of | C2×#30198, #6433735, #6433738, … |
| maximal quotient of | #380119, #380120, #380529, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23186);