Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23387 |
| 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 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
331 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 331 conjugacy classes of subgroups — too many to draw. Only its 135 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:
(1,2,5,8)(3,7,4,6)(9,10,13,16)(11,15,12,14)(17,18)(19,22)(20,24)(21,23)(25,29)(26,31)(27,30)(28,32);(1,3,5,4)(2,6,8,7)(9,11,13,12)(10,14,16,15)(17,19)(18,22)(20,25)(21,27)(23,30)(24,29)(26,28)(31,32);(1,3,5,4)(2,7,8,6)(9,12,13,11)(10,15,16,14)(19,25)(20,28)(22,29)(24,32)(26,27)(30,31);(1,4,5,3)(2,6,8,7)(18,23)(22,30)(24,32)(29,31)Polycyclic presentation:
2^8|a^2=e,b^2=e,c^2=e,d^2=ef,g^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #144587, #144588, C2×#23842, … |
| maximal normal subgroup of | #144587, #144588, C2×#23842, … |
| maximal quotient of | #358733, #358734, #358737, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16830);