Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32712 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
306 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 306 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:
(1,2,4,6)(3,5,7,8)(9,10)(11,16)(12,14)(13,15)(17,18)(19,25)(20,22)(21,24)(23,27)(26,31)(28,30)(29,32);(1,3)(2,5)(4,7)(6,8)(9,11,13,12)(10,14,15,16)(18,22)(19,26)(21,28)(23,31)(25,32)(27,29);(2,6)(5,8)(10,15)(11,12)(17,19)(18,23)(20,27)(21,29)(22,25)(24,31)(26,28)(30,32);(3,7)(5,8)(9,12,13,11)(10,14,15,16)(18,22)(19,27)(24,30)(26,29)Polycyclic presentation:
2^8|a^2=ef,b^2=eh,d^2=eh,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.
| maximal subgroup of | C2×#32830, #6631520, #6631527, … |
| maximal normal subgroup of | C2×#32830, #6631520, #6631527, … |
| maximal quotient of | #391299, #391336, #391369, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25818);