Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13389 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2033 |
| 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 | |
201 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 201 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)(5,18)(9,24)(10,27)(11,16)(13,31)(14,19)(15,30)(17,20)(21,29)(22,25)(26,28);(1,2)(3,13)(4,16)(5,10)(6,8)(7,11)(9,15)(12,28)(14,22)(17,32)(18,23)(19,31)(20,27)(21,26)(24,29)(25,30);(1,3,4,14,7,15,6,21)(2,9,8,22,11,13,16,26)(5,19,17,12,20,29,18,30)(10,24,23,28,27,31,32,25)Polycyclic presentation:
2^8|c^2=d,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^c=egh,f^c=fh,g^c=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13389, #499025, #499603, … |
| maximal normal subgroup of | C2×#13389, #499025, #499603, … |
| maximal quotient of | #144686, #144778, #145002, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6377);