Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13402 |
| 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 | |
202 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 202 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)(3,13)(4,6)(5,19)(9,24)(10,27)(11,12)(14,32)(15,20)(16,30)(17,31)(18,21)(22,25)(26,28);(1,2)(3,14)(4,11)(5,10)(6,8)(7,12)(9,15)(13,28)(16,26)(17,22)(18,29)(19,23)(20,32)(21,27)(24,31)(25,30);(1,3)(2,9,12,26)(4,16)(5,20)(6,15)(7,17)(8,22,11,14)(10,28,27,24)(13,19)(18,31)(21,30)(23,25,29,32)Polycyclic presentation:
2^8|c^2=dg,d^2=h,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eg,f^a=fh,g^a=gh,c^b=cf,d^b=dh,f^b=fh,d^c=dh,e^c=eg,g^c=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13402, #499040, #499514, … |
| maximal normal subgroup of | C2×#13402, #499040, #499514, … |
| maximal quotient of | #144720, #144748, #145027, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6390);