Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13404 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2034 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#472 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
203 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 203 conjugacy classes of subgroups — too many to draw. Only its 53 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) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,8)(3,12)(4,6)(5,19)(9,25)(10,20)(11,17)(13,28)(14,21)(15,30)(16,29)(18,22)(23,27)(26,31);(1,2)(3,13)(4,17)(5,20)(6,8)(7,11)(9,15)(10,22)(12,27)(14,26)(16,23)(18,24)(19,32)(21,28)(25,29)(30,31);(1,3)(2,9,11,13)(4,16)(5,21)(6,14)(7,15)(8,23,17,26)(10,28,20,25)(12,19)(18,30)(22,29)(24,27,32,31)Polycyclic presentation:
2^8|c^2=dg,d^2=h,e^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eg,g^a=gh,c^b=cf,d^b=dh,e^b=eh,d^c=dh,e^c=eg,f^c=fh,g^c=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13404, #499043, #499603, … |
| maximal normal subgroup of | C2×#13404, #499043, #499603, … |
| maximal quotient of | #144718, #144746, #145027, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6392);